Properties

Label 2-4680-1.1-c1-0-18
Degree $2$
Conductor $4680$
Sign $1$
Analytic cond. $37.3699$
Root an. cond. $6.11309$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 3.43·7-s + 2.63·11-s − 13-s − 7.84·17-s + 0.794·19-s + 3.43·23-s + 25-s + 4.06·29-s + 9.27·31-s − 3.43·35-s − 0.636·37-s + 4.63·41-s + 2.41·43-s − 5.27·47-s + 4.77·49-s − 11.4·53-s − 2.63·55-s + 12.1·59-s − 11.4·61-s + 65-s − 6.41·67-s + 10.6·71-s + 16.0·73-s + 9.04·77-s + 10.6·79-s + 16.1·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.29·7-s + 0.795·11-s − 0.277·13-s − 1.90·17-s + 0.182·19-s + 0.715·23-s + 0.200·25-s + 0.755·29-s + 1.66·31-s − 0.580·35-s − 0.104·37-s + 0.724·41-s + 0.367·43-s − 0.769·47-s + 0.682·49-s − 1.57·53-s − 0.355·55-s + 1.57·59-s − 1.47·61-s + 0.124·65-s − 0.783·67-s + 1.26·71-s + 1.88·73-s + 1.03·77-s + 1.19·79-s + 1.77·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4680\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(37.3699\)
Root analytic conductor: \(6.11309\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4680,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.129817105\)
\(L(\frac12)\) \(\approx\) \(2.129817105\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + T \)
13 \( 1 + T \)
good7 \( 1 - 3.43T + 7T^{2} \)
11 \( 1 - 2.63T + 11T^{2} \)
17 \( 1 + 7.84T + 17T^{2} \)
19 \( 1 - 0.794T + 19T^{2} \)
23 \( 1 - 3.43T + 23T^{2} \)
29 \( 1 - 4.06T + 29T^{2} \)
31 \( 1 - 9.27T + 31T^{2} \)
37 \( 1 + 0.636T + 37T^{2} \)
41 \( 1 - 4.63T + 41T^{2} \)
43 \( 1 - 2.41T + 43T^{2} \)
47 \( 1 + 5.27T + 47T^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 - 12.1T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 + 6.41T + 67T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 - 16.0T + 73T^{2} \)
79 \( 1 - 10.6T + 79T^{2} \)
83 \( 1 - 16.1T + 83T^{2} \)
89 \( 1 + 8.63T + 89T^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−8.211922450059555360762372768688, −7.76810558669900142327203018763, −6.71540403167338805470065555144, −6.40890626933003294616995461830, −5.03452532530660049121570644069, −4.66664519377771214813895867618, −3.96871042707624413497712826668, −2.80397745882173519089030859073, −1.89729809055040048663653350488, −0.836348112622736929936553495723, 0.836348112622736929936553495723, 1.89729809055040048663653350488, 2.80397745882173519089030859073, 3.96871042707624413497712826668, 4.66664519377771214813895867618, 5.03452532530660049121570644069, 6.40890626933003294616995461830, 6.71540403167338805470065555144, 7.76810558669900142327203018763, 8.211922450059555360762372768688

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