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Properties

Label 2-10304-1.1-c1-0-15
Degree $2$
Conductor $10304$
Sign $-1$
Analytic cond. $82.2778$
Root an. cond. $9.07071$
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  − 2·5-s − 7-s − 3·9-s + 4·11-s − 6·13-s − 2·17-s + 4·19-s + 23-s − 25-s + 2·29-s + 4·31-s + 2·35-s + 2·37-s − 6·41-s + 12·43-s + 6·45-s + 12·47-s + 49-s + 10·53-s − 8·55-s − 2·61-s + 3·63-s + 12·65-s + 12·67-s − 8·71-s − 14·73-s − 4·77-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s − 9-s + 1.20·11-s − 1.66·13-s − 0.485·17-s + 0.917·19-s + 0.208·23-s − 1/5·25-s + 0.371·29-s + 0.718·31-s + 0.338·35-s + 0.328·37-s − 0.937·41-s + 1.82·43-s + 0.894·45-s + 1.75·47-s + 1/7·49-s + 1.37·53-s − 1.07·55-s − 0.256·61-s + 0.377·63-s + 1.48·65-s + 1.46·67-s − 0.949·71-s − 1.63·73-s − 0.455·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10304\)    =    \(2^{6} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(82.2778\)
Root analytic conductor: \(9.07071\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 10304,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
23 \( 1 - T \)
good3 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
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

−16.90150296082827, −16.36271236126780, −15.68831776921830, −15.15959476909641, −14.62878444397166, −14.05363586564356, −13.64079383249160, −12.61678225461461, −12.12111208993780, −11.71332183629421, −11.34737734770499, −10.41017624247631, −9.810964226073176, −9.086686333952566, −8.740872788338138, −7.818840142833433, −7.318128062555862, −6.757041140177942, −5.920123520271616, −5.268918662138750, −4.371536678424247, −3.890292549674925, −2.924413868513031, −2.414649004096250, −0.9859231765263382, 0, 0.9859231765263382, 2.414649004096250, 2.924413868513031, 3.890292549674925, 4.371536678424247, 5.268918662138750, 5.920123520271616, 6.757041140177942, 7.318128062555862, 7.818840142833433, 8.740872788338138, 9.086686333952566, 9.810964226073176, 10.41017624247631, 11.34737734770499, 11.71332183629421, 12.12111208993780, 12.61678225461461, 13.64079383249160, 14.05363586564356, 14.62878444397166, 15.15959476909641, 15.68831776921830, 16.36271236126780, 16.90150296082827

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