Properties

Label 2-1260-315.34-c0-0-2
Degree $2$
Conductor $1260$
Sign $0.766 + 0.642i$
Analytic cond. $0.628821$
Root an. cond. $0.792982$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)11-s + (−1 − 1.73i)13-s + 0.999·15-s + 2·17-s + (−0.499 − 0.866i)21-s + (−0.499 + 0.866i)25-s + 0.999·27-s + (0.5 − 0.866i)29-s + (0.499 + 0.866i)33-s + 0.999·35-s + 1.99·39-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)11-s + (−1 − 1.73i)13-s + 0.999·15-s + 2·17-s + (−0.499 − 0.866i)21-s + (−0.499 + 0.866i)25-s + 0.999·27-s + (0.5 − 0.866i)29-s + (0.499 + 0.866i)33-s + 0.999·35-s + 1.99·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(0.628821\)
Root analytic conductor: \(0.792982\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :0),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7048396350\)
\(L(\frac12)\) \(\approx\) \(0.7048396350\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - 2T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.893887134316449520003900315330, −9.041205284071703107447268688878, −8.315673376621380181281186083999, −7.54736716130597860415563334494, −6.00862089402620994796524798995, −5.57621709765312417170182597516, −4.86853994502728291245196181917, −3.60742640802207272197799622227, −2.97895615242959328264702728955, −0.70555636344158107069093094353, 1.45142171725875049176399491500, 2.75524952987346086353900111449, 3.91042395338467591995970446413, 4.83677616947501216250468290792, 6.12070785493283817583891144665, 6.98078418398655709014225964889, 7.20111279648771999587577018523, 7.959218325176305146960312070330, 9.348551761298826515776756762890, 10.06730644495767293612875970793

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