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Properties

Label 2-32340-1.1-c1-0-7
Degree $2$
Conductor $32340$
Sign $1$
Analytic cond. $258.236$
Root an. cond. $16.0697$
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 + 5-s + 9-s − 11-s − 15-s − 3·17-s + 19-s − 7·23-s + 25-s − 27-s + 5·29-s + 6·31-s + 33-s + 8·37-s + 2·41-s − 7·43-s + 45-s + 2·47-s + 3·51-s + 5·53-s − 55-s − 57-s + 7·59-s + 7·61-s − 14·67-s + 7·69-s + 2·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.258·15-s − 0.727·17-s + 0.229·19-s − 1.45·23-s + 1/5·25-s − 0.192·27-s + 0.928·29-s + 1.07·31-s + 0.174·33-s + 1.31·37-s + 0.312·41-s − 1.06·43-s + 0.149·45-s + 0.291·47-s + 0.420·51-s + 0.686·53-s − 0.134·55-s − 0.132·57-s + 0.911·59-s + 0.896·61-s − 1.71·67-s + 0.842·69-s + 0.237·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32340\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(258.236\)
Root analytic conductor: \(16.0697\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 32340,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.655539549\)
\(L(\frac12)\) \(\approx\) \(1.655539549\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 \)
11 \( 1 + T \)
good13 \( 1 + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−15.12697758957002, −14.48539360176427, −13.90628538572387, −13.39499420713761, −13.07747629064183, −12.21513558595621, −11.96914726510950, −11.33438305903681, −10.78464532533326, −10.13166932461153, −9.884706297997829, −9.223012037827529, −8.419748606615156, −8.077835389597109, −7.315583002860115, −6.607127580811086, −6.251686116063433, −5.623077835178270, −5.011342787755413, −4.373591497028813, −3.836277137517175, −2.771126290573313, −2.296807929844096, −1.382187273195809, −0.5216251381083001, 0.5216251381083001, 1.382187273195809, 2.296807929844096, 2.771126290573313, 3.836277137517175, 4.373591497028813, 5.011342787755413, 5.623077835178270, 6.251686116063433, 6.607127580811086, 7.315583002860115, 8.077835389597109, 8.419748606615156, 9.223012037827529, 9.884706297997829, 10.13166932461153, 10.78464532533326, 11.33438305903681, 11.96914726510950, 12.21513558595621, 13.07747629064183, 13.39499420713761, 13.90628538572387, 14.48539360176427, 15.12697758957002

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