Properties

Label 2-1260-1.1-c1-0-9
Degree $2$
Conductor $1260$
Sign $-1$
Analytic cond. $10.0611$
Root an. cond. $3.17193$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s − 6·11-s − 4·13-s − 6·17-s + 2·19-s + 25-s − 6·29-s − 10·31-s + 35-s + 2·37-s + 6·41-s − 4·43-s + 49-s + 12·53-s − 6·55-s + 14·61-s − 4·65-s − 4·67-s − 6·71-s − 4·73-s − 6·77-s − 16·79-s + 12·83-s − 6·85-s − 6·89-s − 4·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 1.80·11-s − 1.10·13-s − 1.45·17-s + 0.458·19-s + 1/5·25-s − 1.11·29-s − 1.79·31-s + 0.169·35-s + 0.328·37-s + 0.937·41-s − 0.609·43-s + 1/7·49-s + 1.64·53-s − 0.809·55-s + 1.79·61-s − 0.496·65-s − 0.488·67-s − 0.712·71-s − 0.468·73-s − 0.683·77-s − 1.80·79-s + 1.31·83-s − 0.650·85-s − 0.635·89-s − 0.419·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-1$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1260} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
good11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−9.333092347145761586280629276704, −8.486796104743842391475144783781, −7.52901960668867754280433642901, −7.04398348805248716401259437199, −5.63223394326959203086574118577, −5.21354657812453653302539631933, −4.18698015044674762527792039200, −2.72394457334941735951674528638, −2.03122373506034402447940694337, 0, 2.03122373506034402447940694337, 2.72394457334941735951674528638, 4.18698015044674762527792039200, 5.21354657812453653302539631933, 5.63223394326959203086574118577, 7.04398348805248716401259437199, 7.52901960668867754280433642901, 8.486796104743842391475144783781, 9.333092347145761586280629276704

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