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Properties

Label 2-2912-364.51-c0-0-0
Degree 22
Conductor 29122912
Sign 0.197−0.980i0.197 - 0.980i
Analytic cond. 1.453271.45327
Root an. cond. 1.205511.20551
Motivic weight 00
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 + 0.5i)3-s + (−0.866 − 0.5i)5-s − 7-s + (−0.5 − 0.866i)11-s + 13-s + 0.999·15-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.866 − 0.5i)21-s + (−0.866 − 0.5i)23-s − i·27-s + (0.5 + 0.866i)31-s + (0.866 + 0.499i)33-s + (0.866 + 0.5i)35-s + (−0.866 − 0.5i)37-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (−0.866 − 0.5i)5-s − 7-s + (−0.5 − 0.866i)11-s + 13-s + 0.999·15-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.866 − 0.5i)21-s + (−0.866 − 0.5i)23-s − i·27-s + (0.5 + 0.866i)31-s + (0.866 + 0.499i)33-s + (0.866 + 0.5i)35-s + (−0.866 − 0.5i)37-s + ⋯

Functional equation

Λ(s)=(2912s/2 ΓC(s) L(s)=((0.197−0.980i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2912s/2 ΓC(s) L(s)=((0.197−0.980i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 29122912    =    25⋅7⋅132^{5} \cdot 7 \cdot 13
Sign: 0.197−0.980i0.197 - 0.980i
Analytic conductor: 1.453271.45327
Root analytic conductor: 1.205511.20551
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2912(415,⋅)\chi_{2912} (415, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2912, ( :0), 0.197−0.980i)(2,\ 2912,\ (\ :0),\ 0.197 - 0.980i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.45896194860.4589619486
L(12)L(\frac12) ≈\approx 0.45896194860.4589619486
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
7 1+T 1 + T
13 1−T 1 - T
good3 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
5 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
11 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
17 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
19 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
23 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
29 1+T2 1 + T^{2}
31 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
37 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
41 1−2iT−T2 1 - 2iT - T^{2}
43 1−2iT−T2 1 - 2iT - T^{2}
47 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
53 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
59 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
61 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
67 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
71 1+T2 1 + T^{2}
73 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
79 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
83 1+T2 1 + T^{2}
89 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
97 1−T2 1 - T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−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

−9.074194690380133340639334076787, −8.224885753134341190203907937793, −7.894617960890736863506208435254, −6.46728663337270782006890983262, −6.14412316573716145055105993094, −5.24779949924084907972796422112, −4.45806466071811924043723833379, −3.63824258602976444574553503370, −2.85269796237646499722716536620, −0.953532756705708350182267443491, 0.41973649853521409597449471018, 1.99767277119106943672989819978, 3.47333933463075316706528962319, 3.67707373399344166668005155508, 5.13897002112771356812991167224, 5.81803174434588399968681936424, 6.54729330112640457409641588069, 7.26258637062818815694778718031, 7.69581834463195570291595552568, 8.742181666296059642912779016340

Graph of the ZZ-function along the critical line