Properties

Label 2-3920-1.1-c1-0-44
Degree $2$
Conductor $3920$
Sign $-1$
Analytic cond. $31.3013$
Root an. cond. $5.59476$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.87·3-s − 5-s + 0.534·9-s + 3.29·11-s − 4.19·13-s + 1.87·15-s + 1.43·17-s + 1.24·19-s + 0.272·23-s + 25-s + 4.63·27-s − 2.36·29-s − 3.72·31-s − 6.19·33-s + 0.169·37-s + 7.88·39-s − 11.6·41-s + 10.1·43-s − 0.534·45-s + 3.12·47-s − 2.70·51-s + 9.24·53-s − 3.29·55-s − 2.33·57-s + 9.07·59-s − 7.27·61-s + 4.19·65-s + ⋯
L(s)  = 1  − 1.08·3-s − 0.447·5-s + 0.178·9-s + 0.993·11-s − 1.16·13-s + 0.485·15-s + 0.348·17-s + 0.285·19-s + 0.0568·23-s + 0.200·25-s + 0.892·27-s − 0.438·29-s − 0.669·31-s − 1.07·33-s + 0.0279·37-s + 1.26·39-s − 1.82·41-s + 1.54·43-s − 0.0796·45-s + 0.455·47-s − 0.378·51-s + 1.27·53-s − 0.444·55-s − 0.309·57-s + 1.18·59-s − 0.930·61-s + 0.520·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(31.3013\)
Root analytic conductor: \(5.59476\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3920,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good3 \( 1 + 1.87T + 3T^{2} \)
11 \( 1 - 3.29T + 11T^{2} \)
13 \( 1 + 4.19T + 13T^{2} \)
17 \( 1 - 1.43T + 17T^{2} \)
19 \( 1 - 1.24T + 19T^{2} \)
23 \( 1 - 0.272T + 23T^{2} \)
29 \( 1 + 2.36T + 29T^{2} \)
31 \( 1 + 3.72T + 31T^{2} \)
37 \( 1 - 0.169T + 37T^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 - 10.1T + 43T^{2} \)
47 \( 1 - 3.12T + 47T^{2} \)
53 \( 1 - 9.24T + 53T^{2} \)
59 \( 1 - 9.07T + 59T^{2} \)
61 \( 1 + 7.27T + 61T^{2} \)
67 \( 1 - 13.1T + 67T^{2} \)
71 \( 1 + 6.87T + 71T^{2} \)
73 \( 1 - 15.2T + 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 - 0.167T + 83T^{2} \)
89 \( 1 + 3.09T + 89T^{2} \)
97 \( 1 + 6.60T + 97T^{2} \)
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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

−8.028819918690695941609529734234, −7.10639344532647853670593264539, −6.76981382250095929392043466656, −5.70451326424039637634656107966, −5.26702953586462059524889202200, −4.36627858714822039759696287833, −3.58839845121882205371791950649, −2.46014252882774739442021277849, −1.14249421162625901878574292394, 0, 1.14249421162625901878574292394, 2.46014252882774739442021277849, 3.58839845121882205371791950649, 4.36627858714822039759696287833, 5.26702953586462059524889202200, 5.70451326424039637634656107966, 6.76981382250095929392043466656, 7.10639344532647853670593264539, 8.028819918690695941609529734234

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