Properties

Label 2-1960-1.1-c3-0-67
Degree $2$
Conductor $1960$
Sign $1$
Analytic cond. $115.643$
Root an. cond. $10.7537$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7·3-s − 5·5-s + 22·9-s + 58·11-s + 82·13-s − 35·15-s + 50·17-s + 64·19-s − 111·23-s + 25·25-s − 35·27-s + 103·29-s − 130·31-s + 406·33-s + 376·37-s + 574·39-s − 307·41-s − 197·43-s − 110·45-s + 120·47-s + 350·51-s − 508·53-s − 290·55-s + 448·57-s + 600·59-s − 165·61-s − 410·65-s + ⋯
L(s)  = 1  + 1.34·3-s − 0.447·5-s + 0.814·9-s + 1.58·11-s + 1.74·13-s − 0.602·15-s + 0.713·17-s + 0.772·19-s − 1.00·23-s + 1/5·25-s − 0.249·27-s + 0.659·29-s − 0.753·31-s + 2.14·33-s + 1.67·37-s + 2.35·39-s − 1.16·41-s − 0.698·43-s − 0.364·45-s + 0.372·47-s + 0.960·51-s − 1.31·53-s − 0.710·55-s + 1.04·57-s + 1.32·59-s − 0.346·61-s − 0.782·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(115.643\)
Root analytic conductor: \(10.7537\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.469284624\)
\(L(\frac12)\) \(\approx\) \(4.469284624\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
7 \( 1 \)
good3 \( 1 - 7 T + p^{3} T^{2} \)
11 \( 1 - 58 T + p^{3} T^{2} \)
13 \( 1 - 82 T + p^{3} T^{2} \)
17 \( 1 - 50 T + p^{3} T^{2} \)
19 \( 1 - 64 T + p^{3} T^{2} \)
23 \( 1 + 111 T + p^{3} T^{2} \)
29 \( 1 - 103 T + p^{3} T^{2} \)
31 \( 1 + 130 T + p^{3} T^{2} \)
37 \( 1 - 376 T + p^{3} T^{2} \)
41 \( 1 + 307 T + p^{3} T^{2} \)
43 \( 1 + 197 T + p^{3} T^{2} \)
47 \( 1 - 120 T + p^{3} T^{2} \)
53 \( 1 + 508 T + p^{3} T^{2} \)
59 \( 1 - 600 T + p^{3} T^{2} \)
61 \( 1 + 165 T + p^{3} T^{2} \)
67 \( 1 + 633 T + p^{3} T^{2} \)
71 \( 1 - 840 T + p^{3} T^{2} \)
73 \( 1 - 606 T + p^{3} T^{2} \)
79 \( 1 + 1316 T + p^{3} T^{2} \)
83 \( 1 - 61 T + p^{3} T^{2} \)
89 \( 1 + 187 T + p^{3} T^{2} \)
97 \( 1 - 406 T + p^{3} T^{2} \)
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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

−8.676553474377649942673471848582, −8.247943147732702306971645742872, −7.47996183666778713478632475083, −6.52630658385393580761970481981, −5.79118231618208177825124364375, −4.35695642566528996713525636995, −3.61951646702961242253018422200, −3.22140876217819600507664522674, −1.81462880685168418855658246718, −0.998052417694111841043203596599, 0.998052417694111841043203596599, 1.81462880685168418855658246718, 3.22140876217819600507664522674, 3.61951646702961242253018422200, 4.35695642566528996713525636995, 5.79118231618208177825124364375, 6.52630658385393580761970481981, 7.47996183666778713478632475083, 8.247943147732702306971645742872, 8.676553474377649942673471848582

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