Properties

Label 2-10304-1.1-c1-0-22
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  − 2·3-s + 2·5-s + 7-s + 9-s − 6·11-s + 4·13-s − 4·15-s − 2·17-s − 4·19-s − 2·21-s + 23-s − 25-s + 4·27-s + 10·29-s − 8·31-s + 12·33-s + 2·35-s + 8·37-s − 8·39-s − 2·41-s − 6·43-s + 2·45-s + 12·47-s + 49-s + 4·51-s − 12·53-s − 12·55-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.80·11-s + 1.10·13-s − 1.03·15-s − 0.485·17-s − 0.917·19-s − 0.436·21-s + 0.208·23-s − 1/5·25-s + 0.769·27-s + 1.85·29-s − 1.43·31-s + 2.08·33-s + 0.338·35-s + 1.31·37-s − 1.28·39-s − 0.312·41-s − 0.914·43-s + 0.298·45-s + 1.75·47-s + 1/7·49-s + 0.560·51-s − 1.64·53-s − 1.61·55-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: $\chi_{10304} (1, \cdot )$
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 \( 1 + 2 T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−17.11394405060462, −16.22555451697840, −15.93867934292886, −15.33993118304071, −14.58854008453860, −13.82958758377130, −13.40904527912601, −12.80858750541277, −12.38958666565260, −11.45761352750044, −10.90790669717707, −10.69317930655354, −10.06770770456393, −9.323795413567404, −8.344524364265233, −8.192634383427401, −7.070996025013363, −6.400852131753493, −5.902556694820503, −5.311810109989738, −4.857571092303671, −3.966950946936078, −2.769783455205004, −2.159168990959696, −1.079023234392304, 0, 1.079023234392304, 2.159168990959696, 2.769783455205004, 3.966950946936078, 4.857571092303671, 5.311810109989738, 5.902556694820503, 6.400852131753493, 7.070996025013363, 8.192634383427401, 8.344524364265233, 9.323795413567404, 10.06770770456393, 10.69317930655354, 10.90790669717707, 11.45761352750044, 12.38958666565260, 12.80858750541277, 13.40904527912601, 13.82958758377130, 14.58854008453860, 15.33993118304071, 15.93867934292886, 16.22555451697840, 17.11394405060462

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