Properties

Label 4-28e4-1.1-c1e2-0-41
Degree $4$
Conductor $614656$
Sign $1$
Analytic cond. $39.1909$
Root an. cond. $2.50205$
Motivic weight $1$
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  + 2·3-s + 4·5-s + 3·9-s + 8·15-s + 2·17-s − 2·19-s + 8·23-s + 5·25-s + 10·27-s + 4·29-s + 4·31-s + 6·37-s − 4·41-s − 16·43-s + 12·45-s − 4·47-s + 4·51-s + 10·53-s − 4·57-s + 6·59-s − 4·61-s − 12·67-s + 16·69-s + 14·73-s + 10·75-s − 8·79-s + 20·81-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.78·5-s + 9-s + 2.06·15-s + 0.485·17-s − 0.458·19-s + 1.66·23-s + 25-s + 1.92·27-s + 0.742·29-s + 0.718·31-s + 0.986·37-s − 0.624·41-s − 2.43·43-s + 1.78·45-s − 0.583·47-s + 0.560·51-s + 1.37·53-s − 0.529·57-s + 0.781·59-s − 0.512·61-s − 1.46·67-s + 1.92·69-s + 1.63·73-s + 1.15·75-s − 0.900·79-s + 20/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(614656\)    =    \(2^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(39.1909\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 614656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.876525938\)
\(L(\frac12)\) \(\approx\) \(4.876525938\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−10.18757079550393296912608702903, −10.07146420874460009865246895887, −9.614538880047030733502274423082, −9.385629454929749075372839167549, −8.703615828575611585298132826406, −8.474705804600539351787323355993, −8.234528934786063440154860140464, −7.48875275460665815851260761246, −6.92599685347346531152144476027, −6.64662886532235588787612337134, −6.25629738510967164543363407084, −5.59595534081014243340137400865, −5.12459784446540046217163211228, −4.74651553921977127490508082889, −4.08022324019067241610113110243, −3.34644279965749150319468135345, −2.68402778144868303550679469172, −2.61132341614591274320935923971, −1.62418597665957358110667484267, −1.22054487758578256379095422854, 1.22054487758578256379095422854, 1.62418597665957358110667484267, 2.61132341614591274320935923971, 2.68402778144868303550679469172, 3.34644279965749150319468135345, 4.08022324019067241610113110243, 4.74651553921977127490508082889, 5.12459784446540046217163211228, 5.59595534081014243340137400865, 6.25629738510967164543363407084, 6.64662886532235588787612337134, 6.92599685347346531152144476027, 7.48875275460665815851260761246, 8.234528934786063440154860140464, 8.474705804600539351787323355993, 8.703615828575611585298132826406, 9.385629454929749075372839167549, 9.614538880047030733502274423082, 10.07146420874460009865246895887, 10.18757079550393296912608702903

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