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Properties

Label 2-1260-45.4-c1-0-14
Degree $2$
Conductor $1260$
Sign $0.958 + 0.285i$
Analytic cond. $10.0611$
Root an. cond. $3.17193$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 1.5i)3-s + (−1.86 + 1.23i)5-s + (0.866 − 0.5i)7-s + (−1.5 − 2.59i)9-s + (−2 − 3.46i)11-s + (2.59 + 1.5i)13-s + (−0.232 − 3.86i)15-s + i·17-s + 1.73i·21-s + (−1.73 − i)23-s + (1.96 − 4.59i)25-s + 5.19·27-s + (−4.5 − 7.79i)29-s + (1 − 1.73i)31-s + 6.92·33-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)3-s + (−0.834 + 0.550i)5-s + (0.327 − 0.188i)7-s + (−0.5 − 0.866i)9-s + (−0.603 − 1.04i)11-s + (0.720 + 0.416i)13-s + (−0.0599 − 0.998i)15-s + 0.242i·17-s + 0.377i·21-s + (−0.361 − 0.208i)23-s + (0.392 − 0.919i)25-s + 1.00·27-s + (−0.835 − 1.44i)29-s + (0.179 − 0.311i)31-s + 1.20·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.958 + 0.285i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ 0.958 + 0.285i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9331285820\)
\(L(\frac12)\) \(\approx\) \(0.9331285820\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 1.5i)T \)
5 \( 1 + (1.86 - 1.23i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
good11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.59 - 1.5i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (1.73 + i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + (-4 + 6.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.46 + 2i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.92 + 4i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 + (-1 + 1.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.92 - 4i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 15T + 71T^{2} \)
73 \( 1 - iT - 73T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.33 + 2.5i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + (-8.66 + 5i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.806163210741003495408582665271, −8.756776347163118706167895106360, −8.155604445094922629608977350077, −7.20193056129773641842098672708, −6.12679294336890097457756204427, −5.53703103319753871263825235567, −4.19442792498723532775955895236, −3.83295036199417631668098564157, −2.63792625550613652716420046502, −0.52371962766200080314849693521, 1.03874758617165034622041014301, 2.24260433948137023612694767597, 3.63029367988947763109810121913, 4.86776518831482652007632499713, 5.37291433655454752979645746624, 6.51808465457385390205990271300, 7.43726100580666898065327357964, 7.923735694295337812558084662541, 8.676667305608956921794315144006, 9.681294383027902263978577291799

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