Properties

Label 2-10304-1.1-c1-0-31
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  + 3-s + 2·5-s + 7-s − 2·9-s − 2·11-s + 13-s + 2·15-s − 6·19-s + 21-s − 23-s − 25-s − 5·27-s − 29-s − 31-s − 2·33-s + 2·35-s + 6·37-s + 39-s + 3·41-s − 4·45-s + 3·47-s + 49-s − 6·53-s − 4·55-s − 6·57-s + 8·59-s + 10·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 0.377·7-s − 2/3·9-s − 0.603·11-s + 0.277·13-s + 0.516·15-s − 1.37·19-s + 0.218·21-s − 0.208·23-s − 1/5·25-s − 0.962·27-s − 0.185·29-s − 0.179·31-s − 0.348·33-s + 0.338·35-s + 0.986·37-s + 0.160·39-s + 0.468·41-s − 0.596·45-s + 0.437·47-s + 1/7·49-s − 0.824·53-s − 0.539·55-s − 0.794·57-s + 1.04·59-s + 1.28·61-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 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−16.95031281518099, −16.38836345127907, −15.62989851644835, −15.06847388342920, −14.40743947923022, −14.21138818800045, −13.35145948332259, −13.09757705783944, −12.43590997857899, −11.44767838883151, −11.15268148625401, −10.32352457846236, −9.877402445936206, −9.112038042402011, −8.594154865826952, −8.086246525088619, −7.418803721684858, −6.514423905777972, −5.837492507342590, −5.456708058584579, −4.461249718877446, −3.795292666071134, −2.701838170924305, −2.330923084869619, −1.449661190746120, 0, 1.449661190746120, 2.330923084869619, 2.701838170924305, 3.795292666071134, 4.461249718877446, 5.456708058584579, 5.837492507342590, 6.514423905777972, 7.418803721684858, 8.086246525088619, 8.594154865826952, 9.112038042402011, 9.877402445936206, 10.32352457846236, 11.15268148625401, 11.44767838883151, 12.43590997857899, 13.09757705783944, 13.35145948332259, 14.21138818800045, 14.40743947923022, 15.06847388342920, 15.62989851644835, 16.38836345127907, 16.95031281518099

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