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Properties

Label 2-305760-1.1-c1-0-94
Degree $2$
Conductor $305760$
Sign $-1$
Analytic cond. $2441.50$
Root an. cond. $49.4115$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 9-s + 13-s − 15-s − 2·17-s − 4·19-s − 4·23-s + 25-s − 27-s − 6·29-s + 8·31-s − 6·37-s − 39-s + 2·41-s − 4·43-s + 45-s + 2·51-s + 6·53-s + 4·57-s + 2·61-s + 65-s − 8·67-s + 4·69-s − 6·73-s − 75-s + 4·79-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.277·13-s − 0.258·15-s − 0.485·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.986·37-s − 0.160·39-s + 0.312·41-s − 0.609·43-s + 0.149·45-s + 0.280·51-s + 0.824·53-s + 0.529·57-s + 0.256·61-s + 0.124·65-s − 0.977·67-s + 0.481·69-s − 0.702·73-s − 0.115·75-s + 0.450·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 305760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 305760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(305760\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(2441.50\)
Root analytic conductor: \(49.4115\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 305760,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 \)
13 \( 1 - T \)
good11 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.96872423531842, −12.41425350265435, −11.94374366344767, −11.53987378303749, −11.09883829178566, −10.57106701246942, −10.15722497688341, −9.904692937839348, −9.191843325034315, −8.748025222595027, −8.386331621757964, −7.769325892066569, −7.237466116180231, −6.704586776517987, −6.327541538055610, −5.835669583926488, −5.469305428692885, −4.794061891784046, −4.343245163264591, −3.898884657989216, −3.223157412151129, −2.555934987100444, −1.943672903786312, −1.542127611826897, −0.6763875030827105, 0, 0.6763875030827105, 1.542127611826897, 1.943672903786312, 2.555934987100444, 3.223157412151129, 3.898884657989216, 4.343245163264591, 4.794061891784046, 5.469305428692885, 5.835669583926488, 6.327541538055610, 6.704586776517987, 7.237466116180231, 7.769325892066569, 8.386331621757964, 8.748025222595027, 9.191843325034315, 9.904692937839348, 10.15722497688341, 10.57106701246942, 11.09883829178566, 11.53987378303749, 11.94374366344767, 12.41425350265435, 12.96872423531842

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