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Properties

Label 4-18e2-1.1-c37e2-0-2
Degree $4$
Conductor $324$
Sign $1$
Analytic cond. $24362.6$
Root an. cond. $12.4934$
Motivic weight $37$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.24e5·2-s + 2.06e11·4-s + 1.35e13·5-s + 3.10e15·7-s + 7.20e16·8-s + 7.08e18·10-s − 4.00e18·11-s − 4.17e20·13-s + 1.62e21·14-s + 2.36e22·16-s − 9.24e20·17-s − 1.50e24·19-s + 2.78e24·20-s − 2.09e24·22-s − 4.97e24·23-s + 2.04e24·25-s − 2.19e26·26-s + 6.40e26·28-s − 1.17e27·29-s − 5.52e27·31-s + 7.42e27·32-s − 4.84e26·34-s + 4.19e28·35-s − 1.15e29·37-s − 7.87e29·38-s + 9.73e29·40-s − 4.79e29·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.58·5-s + 0.720·7-s + 1.41·8-s + 2.23·10-s − 0.217·11-s − 1.03·13-s + 1.01·14-s + 5/4·16-s − 0.0159·17-s − 3.31·19-s + 2.37·20-s − 0.306·22-s − 0.319·23-s + 0.0280·25-s − 1.45·26-s + 1.08·28-s − 1.03·29-s − 1.42·31-s + 1.06·32-s − 0.0225·34-s + 1.14·35-s − 1.12·37-s − 4.68·38-s + 2.23·40-s − 0.699·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+37/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(24362.6\)
Root analytic conductor: \(12.4934\)
Motivic weight: \(37\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 324,\ (\ :37/2, 37/2),\ 1)\)

Particular Values

\(L(19)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{39}{2})\) 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$C_1$ \( ( 1 - p^{18} T )^{2} \)
3 \( 1 \)
good5$D_{4}$ \( 1 - 2701506111108 p T + \)\(57\!\cdots\!58\)\( p^{5} T^{2} - 2701506111108 p^{38} T^{3} + p^{74} T^{4} \)
7$D_{4}$ \( 1 - 3106174162962256 T + \)\(46\!\cdots\!86\)\( p^{3} T^{2} - 3106174162962256 p^{37} T^{3} + p^{74} T^{4} \)
11$D_{4}$ \( 1 + 33081641830917624 p^{2} T + \)\(66\!\cdots\!46\)\( p^{5} T^{2} + 33081641830917624 p^{39} T^{3} + p^{74} T^{4} \)
13$D_{4}$ \( 1 + \)\(41\!\cdots\!72\)\( T + \)\(28\!\cdots\!74\)\( p T^{2} + \)\(41\!\cdots\!72\)\( p^{37} T^{3} + p^{74} T^{4} \)
17$D_{4}$ \( 1 + 54405206559310289508 p T + \)\(38\!\cdots\!06\)\( p^{3} T^{2} + 54405206559310289508 p^{38} T^{3} + p^{74} T^{4} \)
19$D_{4}$ \( 1 + \)\(15\!\cdots\!00\)\( T + \)\(51\!\cdots\!62\)\( p T^{2} + \)\(15\!\cdots\!00\)\( p^{37} T^{3} + p^{74} T^{4} \)
23$D_{4}$ \( 1 + \)\(49\!\cdots\!08\)\( T + \)\(13\!\cdots\!14\)\( p T^{2} + \)\(49\!\cdots\!08\)\( p^{37} T^{3} + p^{74} T^{4} \)
29$D_{4}$ \( 1 + \)\(11\!\cdots\!80\)\( T + \)\(76\!\cdots\!42\)\( p T^{2} + \)\(11\!\cdots\!80\)\( p^{37} T^{3} + p^{74} T^{4} \)
31$D_{4}$ \( 1 + \)\(55\!\cdots\!56\)\( T + \)\(12\!\cdots\!26\)\( p T^{2} + \)\(55\!\cdots\!56\)\( p^{37} T^{3} + p^{74} T^{4} \)
37$D_{4}$ \( 1 + \)\(11\!\cdots\!44\)\( T + \)\(23\!\cdots\!18\)\( T^{2} + \)\(11\!\cdots\!44\)\( p^{37} T^{3} + p^{74} T^{4} \)
41$D_{4}$ \( 1 + \)\(47\!\cdots\!44\)\( T + \)\(89\!\cdots\!46\)\( T^{2} + \)\(47\!\cdots\!44\)\( p^{37} T^{3} + p^{74} T^{4} \)
43$D_{4}$ \( 1 + \)\(31\!\cdots\!52\)\( T + \)\(63\!\cdots\!62\)\( T^{2} + \)\(31\!\cdots\!52\)\( p^{37} T^{3} + p^{74} T^{4} \)
47$D_{4}$ \( 1 - \)\(11\!\cdots\!04\)\( T + \)\(12\!\cdots\!78\)\( T^{2} - \)\(11\!\cdots\!04\)\( p^{37} T^{3} + p^{74} T^{4} \)
53$D_{4}$ \( 1 + \)\(53\!\cdots\!28\)\( T + \)\(13\!\cdots\!22\)\( T^{2} + \)\(53\!\cdots\!28\)\( p^{37} T^{3} + p^{74} T^{4} \)
59$D_{4}$ \( 1 + \)\(58\!\cdots\!60\)\( T + \)\(64\!\cdots\!38\)\( T^{2} + \)\(58\!\cdots\!60\)\( p^{37} T^{3} + p^{74} T^{4} \)
61$D_{4}$ \( 1 - \)\(94\!\cdots\!84\)\( T + \)\(19\!\cdots\!06\)\( T^{2} - \)\(94\!\cdots\!84\)\( p^{37} T^{3} + p^{74} T^{4} \)
67$D_{4}$ \( 1 - \)\(32\!\cdots\!96\)\( T + \)\(76\!\cdots\!58\)\( T^{2} - \)\(32\!\cdots\!96\)\( p^{37} T^{3} + p^{74} T^{4} \)
71$D_{4}$ \( 1 - \)\(36\!\cdots\!76\)\( T + \)\(84\!\cdots\!26\)\( T^{2} - \)\(36\!\cdots\!76\)\( p^{37} T^{3} + p^{74} T^{4} \)
73$D_{4}$ \( 1 + \)\(10\!\cdots\!72\)\( T + \)\(44\!\cdots\!02\)\( T^{2} + \)\(10\!\cdots\!72\)\( p^{37} T^{3} + p^{74} T^{4} \)
79$D_{4}$ \( 1 - \)\(41\!\cdots\!20\)\( T + \)\(22\!\cdots\!18\)\( T^{2} - \)\(41\!\cdots\!20\)\( p^{37} T^{3} + p^{74} T^{4} \)
83$D_{4}$ \( 1 + \)\(17\!\cdots\!88\)\( T + \)\(18\!\cdots\!82\)\( T^{2} + \)\(17\!\cdots\!88\)\( p^{37} T^{3} + p^{74} T^{4} \)
89$D_{4}$ \( 1 - \)\(77\!\cdots\!20\)\( T + \)\(24\!\cdots\!58\)\( T^{2} - \)\(77\!\cdots\!20\)\( p^{37} T^{3} + p^{74} T^{4} \)
97$D_{4}$ \( 1 + \)\(65\!\cdots\!44\)\( T + \)\(45\!\cdots\!58\)\( T^{2} + \)\(65\!\cdots\!44\)\( p^{37} T^{3} + p^{74} T^{4} \)
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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

−11.04375935649429014329769803963, −11.00507308858808014936355164040, −10.09642720920238060427184474599, −9.806429710962159079387872340027, −8.812542747198397220608333043687, −8.261663683744928878012512591399, −7.46828412246509636429547571539, −6.72546848827879591046060394218, −6.33691028548862671332045615248, −5.74429197365891142662579088324, −5.09109009456403377672294178314, −4.96046414988016307851153073774, −3.89900145026814522552474577699, −3.78938958309214363844855151143, −2.45469194952268640319407692101, −2.33606966351004398965471762882, −1.69814832284981277033169220480, −1.64253053188381895894773321612, 0, 0, 1.64253053188381895894773321612, 1.69814832284981277033169220480, 2.33606966351004398965471762882, 2.45469194952268640319407692101, 3.78938958309214363844855151143, 3.89900145026814522552474577699, 4.96046414988016307851153073774, 5.09109009456403377672294178314, 5.74429197365891142662579088324, 6.33691028548862671332045615248, 6.72546848827879591046060394218, 7.46828412246509636429547571539, 8.261663683744928878012512591399, 8.812542747198397220608333043687, 9.806429710962159079387872340027, 10.09642720920238060427184474599, 11.00507308858808014936355164040, 11.04375935649429014329769803963

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