Properties

Label 2-10304-1.1-c1-0-20
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  + 2·3-s + 4·5-s + 7-s + 9-s + 4·11-s − 4·13-s + 8·15-s − 6·17-s + 2·19-s + 2·21-s + 23-s + 11·25-s − 4·27-s + 6·29-s + 8·31-s + 8·33-s + 4·35-s − 10·37-s − 8·39-s + 6·41-s + 12·43-s + 4·45-s + 8·47-s + 49-s − 12·51-s + 2·53-s + 16·55-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.78·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 1.10·13-s + 2.06·15-s − 1.45·17-s + 0.458·19-s + 0.436·21-s + 0.208·23-s + 11/5·25-s − 0.769·27-s + 1.11·29-s + 1.43·31-s + 1.39·33-s + 0.676·35-s − 1.64·37-s − 1.28·39-s + 0.937·41-s + 1.82·43-s + 0.596·45-s + 1.16·47-s + 1/7·49-s − 1.68·51-s + 0.274·53-s + 2.15·55-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: \(0\)
Selberg data: \((2,\ 10304,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.238981048\)
\(L(\frac12)\) \(\approx\) \(5.238981048\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−16.94916166131735, −15.80363151491862, −15.39812798426748, −14.47886158428375, −14.29954859611894, −13.86652191017955, −13.46732589662521, −12.68823486033446, −12.08337992637110, −11.36680773072988, −10.48217142183772, −10.04216481397411, −9.343906207038869, −8.887347873131547, −8.700367340589003, −7.557821015779846, −6.990787752681660, −6.272474935407953, −5.717506796755399, −4.753034264609928, −4.282448275224803, −3.062778595310250, −2.472890569720785, −1.982458075272381, −1.069330501498045, 1.069330501498045, 1.982458075272381, 2.472890569720785, 3.062778595310250, 4.282448275224803, 4.753034264609928, 5.717506796755399, 6.272474935407953, 6.990787752681660, 7.557821015779846, 8.700367340589003, 8.887347873131547, 9.343906207038869, 10.04216481397411, 10.48217142183772, 11.36680773072988, 12.08337992637110, 12.68823486033446, 13.46732589662521, 13.86652191017955, 14.29954859611894, 14.47886158428375, 15.39812798426748, 15.80363151491862, 16.94916166131735

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