Properties

Label 2-10304-1.1-c1-0-4
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 − 7-s − 2·9-s − 6·11-s + 3·13-s − 21-s − 23-s − 5·25-s − 5·27-s + 3·29-s + 7·31-s − 6·33-s − 8·37-s + 3·39-s − 11·41-s + 4·43-s − 47-s + 49-s − 4·53-s + 12·59-s + 6·61-s + 2·63-s + 12·67-s − 69-s + 5·71-s + 15·73-s − 5·75-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s − 2/3·9-s − 1.80·11-s + 0.832·13-s − 0.218·21-s − 0.208·23-s − 25-s − 0.962·27-s + 0.557·29-s + 1.25·31-s − 1.04·33-s − 1.31·37-s + 0.480·39-s − 1.71·41-s + 0.609·43-s − 0.145·47-s + 1/7·49-s − 0.549·53-s + 1.56·59-s + 0.768·61-s + 0.251·63-s + 1.46·67-s − 0.120·69-s + 0.593·71-s + 1.75·73-s − 0.577·75-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

\(L(1)\) \(\approx\) \(1.462595345\)
\(L(\frac12)\) \(\approx\) \(1.462595345\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 - 15 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−16.46075372244699, −15.78105218763882, −15.59095037750608, −15.03545376599608, −14.06464691306476, −13.73122925983907, −13.36041694226449, −12.62943159412359, −12.03290844635416, −11.26717899469159, −10.77692196588289, −9.968126982535109, −9.738051750430025, −8.569735699711905, −8.363037082940696, −7.876178677452212, −6.958785752966776, −6.300973253038285, −5.468989432335999, −5.102769735366424, −3.948655304102723, −3.315666430112258, −2.629547020600000, −1.963176171737799, −0.5198216031977594, 0.5198216031977594, 1.963176171737799, 2.629547020600000, 3.315666430112258, 3.948655304102723, 5.102769735366424, 5.468989432335999, 6.300973253038285, 6.958785752966776, 7.876178677452212, 8.363037082940696, 8.569735699711905, 9.738051750430025, 9.968126982535109, 10.77692196588289, 11.26717899469159, 12.03290844635416, 12.62943159412359, 13.36041694226449, 13.73122925983907, 14.06464691306476, 15.03545376599608, 15.59095037750608, 15.78105218763882, 16.46075372244699

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