Properties

Label 2-1260-45.34-c1-0-10
Degree 22
Conductor 12601260
Sign 0.9580.285i0.958 - 0.285i
Analytic cond. 10.061110.0611
Root an. cond. 3.171933.17193
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(1260s/2ΓC(s)L(s)=((0.9580.285i)Λ(2s)\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}
Λ(s)=(1260s/2ΓC(s+1/2)L(s)=((0.9580.285i)Λ(1s)\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: 22
Conductor: 12601260    =    2232572^{2} \cdot 3^{2} \cdot 5 \cdot 7
Sign: 0.9580.285i0.958 - 0.285i
Analytic conductor: 10.061110.0611
Root analytic conductor: 3.171933.17193
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1260(169,)\chi_{1260} (169, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1260, ( :1/2), 0.9580.285i)(2,\ 1260,\ (\ :1/2),\ 0.958 - 0.285i)

Particular Values

L(1)L(1) \approx 0.93312858200.9331285820
L(12)L(\frac12) \approx 0.93312858200.9331285820
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(0.866+1.5i)T 1 + (0.866 + 1.5i)T
5 1+(1.86+1.23i)T 1 + (1.86 + 1.23i)T
7 1+(0.8660.5i)T 1 + (-0.866 - 0.5i)T
good11 1+(23.46i)T+(5.59.52i)T2 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2}
13 1+(2.59+1.5i)T+(6.511.2i)T2 1 + (-2.59 + 1.5i)T + (6.5 - 11.2i)T^{2}
17 1+iT17T2 1 + iT - 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1+(1.73i)T+(11.519.9i)T2 1 + (1.73 - i)T + (11.5 - 19.9i)T^{2}
29 1+(4.57.79i)T+(14.525.1i)T2 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2}
31 1+(11.73i)T+(15.5+26.8i)T2 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2}
37 1+2iT37T2 1 + 2iT - 37T^{2}
41 1+(46.92i)T+(20.5+35.5i)T2 1 + (-4 - 6.92i)T + (-20.5 + 35.5i)T^{2}
43 1+(3.462i)T+(21.5+37.2i)T2 1 + (-3.46 - 2i)T + (21.5 + 37.2i)T^{2}
47 1+(6.924i)T+(23.5+40.7i)T2 1 + (-6.92 - 4i)T + (23.5 + 40.7i)T^{2}
53 1+12iT53T2 1 + 12iT - 53T^{2}
59 1+(11.73i)T+(29.5+51.0i)T2 1 + (-1 - 1.73i)T + (-29.5 + 51.0i)T^{2}
61 1+(5+8.66i)T+(30.552.8i)T2 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2}
67 1+(6.92+4i)T+(33.558.0i)T2 1 + (-6.92 + 4i)T + (33.5 - 58.0i)T^{2}
71 115T+71T2 1 - 15T + 71T^{2}
73 1+iT73T2 1 + iT - 73T^{2}
79 1+(39.568.4i)T2 1 + (-39.5 - 68.4i)T^{2}
83 1+(4.332.5i)T+(41.5+71.8i)T2 1 + (-4.33 - 2.5i)T + (41.5 + 71.8i)T^{2}
89 1+10T+89T2 1 + 10T + 89T^{2}
97 1+(8.665i)T+(48.5+84.0i)T2 1 + (-8.66 - 5i)T + (48.5 + 84.0i)T^{2}
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   L(s)=p j=12(1αj,pps)1L(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

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

Graph of the ZZ-function along the critical line