Properties

Label 2-10304-1.1-c1-0-7
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 $0$

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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 10304,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.903292499\)
\(L(\frac12)\) \(\approx\) \(1.903292499\)
\(L(\frac{3}{2})\) 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

−16.69739487498136, −16.15704279491517, −15.59639873274711, −14.73643583308358, −14.26679006056994, −13.69235640578993, −13.28007437622685, −12.45362554639928, −11.98057996678066, −11.25638166689724, −10.96310484217827, −10.00339464753138, −9.567203873547474, −9.119572201960376, −8.284880253179866, −7.586881461773473, −6.685480038247013, −6.253162100921862, −5.618110340764693, −5.167858244117217, −4.207133840651601, −3.299703141058737, −2.636217999855838, −1.590640948134176, −0.6926816306774573, 0.6926816306774573, 1.590640948134176, 2.636217999855838, 3.299703141058737, 4.207133840651601, 5.167858244117217, 5.618110340764693, 6.253162100921862, 6.685480038247013, 7.586881461773473, 8.284880253179866, 9.119572201960376, 9.567203873547474, 10.00339464753138, 10.96310484217827, 11.25638166689724, 11.98057996678066, 12.45362554639928, 13.28007437622685, 13.69235640578993, 14.26679006056994, 14.73643583308358, 15.59639873274711, 16.15704279491517, 16.69739487498136

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