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Properties

Label 2-2835-315.293-c0-0-1
Degree 22
Conductor 28352835
Sign 0.801+0.597i0.801 + 0.597i
Analytic cond. 1.414841.41484
Root an. cond. 1.189471.18947
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  + (0.258 − 0.965i)2-s + (0.965 − 0.258i)5-s + (0.5 + 0.866i)7-s + (0.707 − 0.707i)8-s − i·10-s + (−1.22 + 0.707i)11-s + (0.366 + 1.36i)13-s + (0.965 − 0.258i)14-s + (−0.5 − 0.866i)16-s + (−0.707 + 0.707i)17-s + 19-s + (0.366 + 1.36i)22-s + (−0.258 − 0.965i)23-s + (0.866 − 0.499i)25-s + 1.41·26-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (0.965 − 0.258i)5-s + (0.5 + 0.866i)7-s + (0.707 − 0.707i)8-s − i·10-s + (−1.22 + 0.707i)11-s + (0.366 + 1.36i)13-s + (0.965 − 0.258i)14-s + (−0.5 − 0.866i)16-s + (−0.707 + 0.707i)17-s + 19-s + (0.366 + 1.36i)22-s + (−0.258 − 0.965i)23-s + (0.866 − 0.499i)25-s + 1.41·26-s + ⋯

Functional equation

Λ(s)=(2835s/2 ΓC(s) L(s)=((0.801+0.597i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2835 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2835s/2 ΓC(s) L(s)=((0.801+0.597i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2835 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 28352835    =    34⋅5⋅73^{4} \cdot 5 \cdot 7
Sign: 0.801+0.597i0.801 + 0.597i
Analytic conductor: 1.414841.41484
Root analytic conductor: 1.189471.18947
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2835(2078,⋅)\chi_{2835} (2078, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2835, ( :0), 0.801+0.597i)(2,\ 2835,\ (\ :0),\ 0.801 + 0.597i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.9291001861.929100186
L(12)L(\frac12) ≈\approx 1.9291001861.929100186
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)
bad3 1 1
5 1+(−0.965+0.258i)T 1 + (-0.965 + 0.258i)T
7 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
good2 1+(−0.258+0.965i)T+(−0.866−0.5i)T2 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2}
11 1+(1.22−0.707i)T+(0.5−0.866i)T2 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2}
13 1+(−0.366−1.36i)T+(−0.866+0.5i)T2 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2}
17 1+(0.707−0.707i)T−iT2 1 + (0.707 - 0.707i)T - iT^{2}
19 1−T+T2 1 - T + T^{2}
23 1+(0.258+0.965i)T+(−0.866+0.5i)T2 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2}
29 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
31 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
37 1−iT2 1 - iT^{2}
41 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
43 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
47 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
53 1+(−0.707+0.707i)T−iT2 1 + (-0.707 + 0.707i)T - iT^{2}
59 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
61 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
67 1+(0.366+1.36i)T+(−0.866+0.5i)T2 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2}
71 1−T2 1 - T^{2}
73 1−iT2 1 - iT^{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+(−0.258+0.965i)T+(−0.866−0.5i)T2 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2}
89 1−1.41iT−T2 1 - 1.41iT - T^{2}
97 1+(−0.366+1.36i)T+(−0.866−0.5i)T2 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)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

−9.091842448431212075022971356729, −8.310967415000583870660116940363, −7.36638860826879873365893506291, −6.54113084286608228679185775503, −5.69052233363673209598990398611, −4.83843405493955903263452960499, −4.20519399287599994929780440874, −2.89268367864770540438659283479, −2.10432090071041048414089705210, −1.69067916094587954425347107375, 1.25560904933219782135934104531, 2.50814225930910585847458600436, 3.39986208141036299490156512662, 4.79588706888379948779018938300, 5.48474827930793682601758413556, 5.76041998054564081019965803431, 6.84216997709932410841533116849, 7.49513336554350351685245673434, 7.950406214537544147540246496202, 8.851214569993272235900063209583

Graph of the ZZ-function along the critical line