Properties

Label 2-10304-1.1-c1-0-12
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 + 2·5-s − 7-s + 9-s + 2·11-s + 4·13-s + 4·15-s − 6·17-s − 2·21-s + 23-s − 25-s − 4·27-s + 2·29-s + 4·31-s + 4·33-s − 2·35-s + 8·39-s + 6·41-s − 6·43-s + 2·45-s + 49-s − 12·51-s + 12·53-s + 4·55-s + 10·59-s − 2·61-s − 63-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 1.10·13-s + 1.03·15-s − 1.45·17-s − 0.436·21-s + 0.208·23-s − 1/5·25-s − 0.769·27-s + 0.371·29-s + 0.718·31-s + 0.696·33-s − 0.338·35-s + 1.28·39-s + 0.937·41-s − 0.914·43-s + 0.298·45-s + 1/7·49-s − 1.68·51-s + 1.64·53-s + 0.539·55-s + 1.30·59-s − 0.256·61-s − 0.125·63-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

\(L(1)\) \(\approx\) \(3.983049561\)
\(L(\frac12)\) \(\approx\) \(3.983049561\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
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

−16.63323598701286, −15.85051099147282, −15.40981109361731, −14.84215360262186, −14.15970383479556, −13.69995624506772, −13.34232818564638, −12.92541280288052, −11.93768486148606, −11.34753122838379, −10.67108696094228, −9.955370439086972, −9.415299875986161, −8.810764761572470, −8.569499206763707, −7.753550245186843, −6.835113692973619, −6.352099786831520, −5.778240533319667, −4.798340526147352, −3.911239940239747, −3.422649850640767, −2.412743225776211, −2.036130209240299, −0.8920875883967132, 0.8920875883967132, 2.036130209240299, 2.412743225776211, 3.422649850640767, 3.911239940239747, 4.798340526147352, 5.778240533319667, 6.352099786831520, 6.835113692973619, 7.753550245186843, 8.569499206763707, 8.810764761572470, 9.415299875986161, 9.955370439086972, 10.67108696094228, 11.34753122838379, 11.93768486148606, 12.92541280288052, 13.34232818564638, 13.69995624506772, 14.15970383479556, 14.84215360262186, 15.40981109361731, 15.85051099147282, 16.63323598701286

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