Properties

Label 2-10304-1.1-c1-0-13
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  + 3·3-s − 2·5-s + 7-s + 6·9-s + 2·11-s + 13-s − 6·15-s + 2·19-s + 3·21-s + 23-s − 25-s + 9·27-s + 3·29-s + 31-s + 6·33-s − 2·35-s + 2·37-s + 3·39-s − 41-s + 8·43-s − 12·45-s + 5·47-s + 49-s + 6·53-s − 4·55-s + 6·57-s − 6·61-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.894·5-s + 0.377·7-s + 2·9-s + 0.603·11-s + 0.277·13-s − 1.54·15-s + 0.458·19-s + 0.654·21-s + 0.208·23-s − 1/5·25-s + 1.73·27-s + 0.557·29-s + 0.179·31-s + 1.04·33-s − 0.338·35-s + 0.328·37-s + 0.480·39-s − 0.156·41-s + 1.21·43-s − 1.78·45-s + 0.729·47-s + 1/7·49-s + 0.824·53-s − 0.539·55-s + 0.794·57-s − 0.768·61-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: \(0\)
Selberg data: \((2,\ 10304,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.224859325\)
\(L(\frac12)\) \(\approx\) \(4.224859325\)
\(L(\frac{3}{2})\) 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 \( 1 - p T + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 5 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−16.45542015515046, −15.66307815476314, −15.45708866271731, −14.87317145166198, −14.28644436431000, −13.86414823102478, −13.38489664823437, −12.56896757450497, −12.06510333057093, −11.41616054020813, −10.72499691281440, −9.965313665953017, −9.336405091107736, −8.772944899944538, −8.346261416828060, −7.619838352199142, −7.373285923132864, −6.515620792262385, −5.549389260815155, −4.398068758558143, −4.134367476137575, −3.345195296057326, −2.748225735279402, −1.840209810219806, −0.9290862462929971, 0.9290862462929971, 1.840209810219806, 2.748225735279402, 3.345195296057326, 4.134367476137575, 4.398068758558143, 5.549389260815155, 6.515620792262385, 7.373285923132864, 7.619838352199142, 8.346261416828060, 8.772944899944538, 9.336405091107736, 9.965313665953017, 10.72499691281440, 11.41616054020813, 12.06510333057093, 12.56896757450497, 13.38489664823437, 13.86414823102478, 14.28644436431000, 14.87317145166198, 15.45708866271731, 15.66307815476314, 16.45542015515046

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