Properties

Label 2-3920-1.1-c1-0-39
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.41·3-s − 5-s + 2.82·9-s + 5.82·11-s − 1.58·13-s − 2.41·15-s + 5.24·17-s + 6·19-s − 4.58·23-s + 25-s − 0.414·27-s + 2.65·29-s + 1.75·31-s + 14.0·33-s − 6.24·37-s − 3.82·39-s − 2.24·41-s − 2·43-s − 2.82·45-s + 1.24·47-s + 12.6·51-s − 4.24·53-s − 5.82·55-s + 14.4·57-s + 6.24·59-s − 2.82·61-s + 1.58·65-s + ⋯
L(s)  = 1  + 1.39·3-s − 0.447·5-s + 0.942·9-s + 1.75·11-s − 0.439·13-s − 0.623·15-s + 1.27·17-s + 1.37·19-s − 0.956·23-s + 0.200·25-s − 0.0797·27-s + 0.493·29-s + 0.315·31-s + 2.44·33-s − 1.02·37-s − 0.613·39-s − 0.350·41-s − 0.304·43-s − 0.421·45-s + 0.181·47-s + 1.77·51-s − 0.582·53-s − 0.785·55-s + 1.91·57-s + 0.812·59-s − 0.362·61-s + 0.196·65-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: \(0\)
Selberg data: \((2,\ 3920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.430381146\)
\(L(\frac12)\) \(\approx\) \(3.430381146\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.41T + 3T^{2} \)
11 \( 1 - 5.82T + 11T^{2} \)
13 \( 1 + 1.58T + 13T^{2} \)
17 \( 1 - 5.24T + 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + 4.58T + 23T^{2} \)
29 \( 1 - 2.65T + 29T^{2} \)
31 \( 1 - 1.75T + 31T^{2} \)
37 \( 1 + 6.24T + 37T^{2} \)
41 \( 1 + 2.24T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 - 1.24T + 47T^{2} \)
53 \( 1 + 4.24T + 53T^{2} \)
59 \( 1 - 6.24T + 59T^{2} \)
61 \( 1 + 2.82T + 61T^{2} \)
67 \( 1 + 0.242T + 67T^{2} \)
71 \( 1 - 8.82T + 71T^{2} \)
73 \( 1 + 8.48T + 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 8T + 89T^{2} \)
97 \( 1 + 4.75T + 97T^{2} \)
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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

−8.441442790812856523883432212045, −7.82713484789237901041498766749, −7.24035620634240993771593864480, −6.44770162826730772498000048947, −5.43603910847013014003027763175, −4.42309441543727187005896293854, −3.51597730171840436274601175836, −3.28239995162409282042144765754, −2.03756948013824219197765158176, −1.07790614661032914947477398655, 1.07790614661032914947477398655, 2.03756948013824219197765158176, 3.28239995162409282042144765754, 3.51597730171840436274601175836, 4.42309441543727187005896293854, 5.43603910847013014003027763175, 6.44770162826730772498000048947, 7.24035620634240993771593864480, 7.82713484789237901041498766749, 8.441442790812856523883432212045

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