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Properties

Label 4-2e8-1.1-c29e2-0-0
Degree $4$
Conductor $256$
Sign $1$
Analytic cond. $7266.68$
Root an. cond. $9.23281$
Motivic weight $29$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.96e6·3-s − 1.74e10·5-s + 3.02e12·7-s + 2.54e13·9-s + 2.05e15·11-s + 1.71e16·13-s − 8.68e16·15-s − 6.64e17·17-s − 1.23e18·19-s + 1.50e19·21-s + 1.85e19·23-s − 1.37e20·25-s + 4.71e20·27-s + 9.96e20·29-s + 1.08e21·31-s + 1.02e22·33-s − 5.27e22·35-s + 9.88e22·37-s + 8.51e22·39-s − 1.06e23·41-s − 5.10e23·43-s − 4.44e23·45-s + 4.52e24·47-s + 2.58e24·49-s − 3.30e24·51-s − 1.61e25·53-s − 3.59e25·55-s + ⋯
L(s)  = 1  + 0.599·3-s − 1.28·5-s + 1.68·7-s + 0.370·9-s + 1.63·11-s + 1.20·13-s − 0.767·15-s − 0.957·17-s − 0.353·19-s + 1.00·21-s + 0.334·23-s − 0.735·25-s + 0.828·27-s + 0.621·29-s + 0.257·31-s + 0.978·33-s − 2.15·35-s + 1.80·37-s + 0.723·39-s − 0.439·41-s − 1.05·43-s − 0.474·45-s + 2.56·47-s + 0.801·49-s − 0.574·51-s − 1.60·53-s − 2.08·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+29/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $1$
Analytic conductor: \(7266.68\)
Root analytic conductor: \(9.23281\)
Motivic weight: \(29\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 256,\ (\ :29/2, 29/2),\ 1)\)

Particular Values

\(L(15)\) \(\approx\) \(6.004082728\)
\(L(\frac12)\) \(\approx\) \(6.004082728\)
\(L(\frac{31}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
good3$D_{4}$ \( 1 - 551960 p^{2} T - 350106910 p^{7} T^{2} - 551960 p^{31} T^{3} + p^{58} T^{4} \)
5$D_{4}$ \( 1 + 139822308 p^{3} T + 5664377441277062 p^{7} T^{2} + 139822308 p^{32} T^{3} + p^{58} T^{4} \)
7$D_{4}$ \( 1 - 431473240400 p T + \)\(19\!\cdots\!50\)\( p^{3} T^{2} - 431473240400 p^{30} T^{3} + p^{58} T^{4} \)
11$D_{4}$ \( 1 - 186852774508056 p T + \)\(31\!\cdots\!66\)\( p^{3} T^{2} - 186852774508056 p^{30} T^{3} + p^{58} T^{4} \)
13$D_{4}$ \( 1 - 1318413167640940 p T + \)\(21\!\cdots\!30\)\( p^{3} T^{2} - 1318413167640940 p^{30} T^{3} + p^{58} T^{4} \)
17$D_{4}$ \( 1 + 39105642818102940 p T + \)\(31\!\cdots\!30\)\( p^{2} T^{2} + 39105642818102940 p^{30} T^{3} + p^{58} T^{4} \)
19$D_{4}$ \( 1 + 1232169445452155080 T + \)\(78\!\cdots\!82\)\( p T^{2} + 1232169445452155080 p^{29} T^{3} + p^{58} T^{4} \)
23$D_{4}$ \( 1 - 808192630624973040 p T + \)\(11\!\cdots\!10\)\( p^{2} T^{2} - 808192630624973040 p^{30} T^{3} + p^{58} T^{4} \)
29$D_{4}$ \( 1 - \)\(99\!\cdots\!20\)\( T + \)\(34\!\cdots\!38\)\( T^{2} - \)\(99\!\cdots\!20\)\( p^{29} T^{3} + p^{58} T^{4} \)
31$D_{4}$ \( 1 - \)\(10\!\cdots\!56\)\( T + \)\(17\!\cdots\!26\)\( T^{2} - \)\(10\!\cdots\!56\)\( p^{29} T^{3} + p^{58} T^{4} \)
37$D_{4}$ \( 1 - \)\(98\!\cdots\!60\)\( T + \)\(75\!\cdots\!10\)\( T^{2} - \)\(98\!\cdots\!60\)\( p^{29} T^{3} + p^{58} T^{4} \)
41$D_{4}$ \( 1 + \)\(10\!\cdots\!76\)\( T + \)\(11\!\cdots\!66\)\( T^{2} + \)\(10\!\cdots\!76\)\( p^{29} T^{3} + p^{58} T^{4} \)
43$D_{4}$ \( 1 + \)\(51\!\cdots\!00\)\( T + \)\(36\!\cdots\!50\)\( T^{2} + \)\(51\!\cdots\!00\)\( p^{29} T^{3} + p^{58} T^{4} \)
47$D_{4}$ \( 1 - \)\(45\!\cdots\!20\)\( T + \)\(11\!\cdots\!30\)\( T^{2} - \)\(45\!\cdots\!20\)\( p^{29} T^{3} + p^{58} T^{4} \)
53$D_{4}$ \( 1 + \)\(16\!\cdots\!40\)\( T + \)\(22\!\cdots\!30\)\( T^{2} + \)\(16\!\cdots\!40\)\( p^{29} T^{3} + p^{58} T^{4} \)
59$D_{4}$ \( 1 - \)\(83\!\cdots\!60\)\( T + \)\(40\!\cdots\!78\)\( T^{2} - \)\(83\!\cdots\!60\)\( p^{29} T^{3} + p^{58} T^{4} \)
61$D_{4}$ \( 1 + \)\(15\!\cdots\!16\)\( T + \)\(90\!\cdots\!46\)\( T^{2} + \)\(15\!\cdots\!16\)\( p^{29} T^{3} + p^{58} T^{4} \)
67$D_{4}$ \( 1 + \)\(24\!\cdots\!20\)\( T + \)\(16\!\cdots\!70\)\( T^{2} + \)\(24\!\cdots\!20\)\( p^{29} T^{3} + p^{58} T^{4} \)
71$D_{4}$ \( 1 - \)\(18\!\cdots\!36\)\( T + \)\(90\!\cdots\!86\)\( T^{2} - \)\(18\!\cdots\!36\)\( p^{29} T^{3} + p^{58} T^{4} \)
73$D_{4}$ \( 1 - \)\(99\!\cdots\!80\)\( T + \)\(15\!\cdots\!90\)\( T^{2} - \)\(99\!\cdots\!80\)\( p^{29} T^{3} + p^{58} T^{4} \)
79$D_{4}$ \( 1 - \)\(72\!\cdots\!80\)\( T + \)\(34\!\cdots\!38\)\( T^{2} - \)\(72\!\cdots\!80\)\( p^{29} T^{3} + p^{58} T^{4} \)
83$D_{4}$ \( 1 + \)\(22\!\cdots\!40\)\( T + \)\(60\!\cdots\!70\)\( T^{2} + \)\(22\!\cdots\!40\)\( p^{29} T^{3} + p^{58} T^{4} \)
89$D_{4}$ \( 1 - \)\(58\!\cdots\!60\)\( T + \)\(67\!\cdots\!18\)\( T^{2} - \)\(58\!\cdots\!60\)\( p^{29} T^{3} + p^{58} T^{4} \)
97$D_{4}$ \( 1 - \)\(10\!\cdots\!80\)\( T + \)\(96\!\cdots\!30\)\( T^{2} - \)\(10\!\cdots\!80\)\( p^{29} T^{3} + p^{58} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.21841116812291819341800708753, −12.32847946220804023322307981389, −11.60464586716373825919453398301, −11.38728223628591102825847138024, −10.89351428462092866346687596770, −9.878582406449349397972837742266, −8.767826764517269843168504643221, −8.763355374010825001777509488740, −7.956930518452058018356224389298, −7.58767197147337532164165426074, −6.62618956029365923702167803278, −6.14352480035345943641031251515, −4.89803425075451737958421294886, −4.38032976923868202807743925792, −3.92638684786458758529699218215, −3.45113957759177797293274179789, −2.32319717779621819210247455238, −1.75857757063001445114377070586, −1.04364877224501468619035505866, −0.63689132156167635277638810765, 0.63689132156167635277638810765, 1.04364877224501468619035505866, 1.75857757063001445114377070586, 2.32319717779621819210247455238, 3.45113957759177797293274179789, 3.92638684786458758529699218215, 4.38032976923868202807743925792, 4.89803425075451737958421294886, 6.14352480035345943641031251515, 6.62618956029365923702167803278, 7.58767197147337532164165426074, 7.956930518452058018356224389298, 8.763355374010825001777509488740, 8.767826764517269843168504643221, 9.878582406449349397972837742266, 10.89351428462092866346687596770, 11.38728223628591102825847138024, 11.60464586716373825919453398301, 12.32847946220804023322307981389, 13.21841116812291819341800708753

Graph of the $Z$-function along the critical line