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Properties

Label 2-1904-476.191-c0-0-1
Degree 22
Conductor 19041904
Sign 0.834+0.550i0.834 + 0.550i
Analytic cond. 0.9502190.950219
Root an. cond. 0.9747920.974792
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)3-s + (0.366 + 1.36i)5-s + (0.707 − 0.707i)7-s + (−0.965 − 0.258i)11-s − 13-s + 1.41·15-s + (0.866 − 0.5i)17-s + (0.707 − 1.22i)19-s + (−0.500 − 0.866i)21-s + (−0.866 + 0.5i)25-s + (0.707 − 0.707i)27-s + (1 − i)29-s + (−0.517 + 1.93i)31-s + (−0.499 + 0.866i)33-s + (1.22 + 0.707i)35-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)3-s + (0.366 + 1.36i)5-s + (0.707 − 0.707i)7-s + (−0.965 − 0.258i)11-s − 13-s + 1.41·15-s + (0.866 − 0.5i)17-s + (0.707 − 1.22i)19-s + (−0.500 − 0.866i)21-s + (−0.866 + 0.5i)25-s + (0.707 − 0.707i)27-s + (1 − i)29-s + (−0.517 + 1.93i)31-s + (−0.499 + 0.866i)33-s + (1.22 + 0.707i)35-s + ⋯

Functional equation

Λ(s)=(1904s/2 ΓC(s) L(s)=((0.834+0.550i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1904s/2 ΓC(s) L(s)=((0.834+0.550i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19041904    =    24⋅7⋅172^{4} \cdot 7 \cdot 17
Sign: 0.834+0.550i0.834 + 0.550i
Analytic conductor: 0.9502190.950219
Root analytic conductor: 0.9747920.974792
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1904(191,⋅)\chi_{1904} (191, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1904, ( :0), 0.834+0.550i)(2,\ 1904,\ (\ :0),\ 0.834 + 0.550i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4218613991.421861399
L(12)L(\frac12) ≈\approx 1.4218613991.421861399
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+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
17 1+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
good3 1+(−0.258+0.965i)T+(−0.866−0.5i)T2 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2}
5 1+(−0.366−1.36i)T+(−0.866+0.5i)T2 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2}
11 1+(0.965+0.258i)T+(0.866+0.5i)T2 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2}
13 1+T+T2 1 + T + T^{2}
19 1+(−0.707+1.22i)T+(−0.5−0.866i)T2 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2}
23 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
29 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
31 1+(0.517−1.93i)T+(−0.866−0.5i)T2 1 + (0.517 - 1.93i)T + (-0.866 - 0.5i)T^{2}
37 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
41 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
43 1+T2 1 + T^{2}
47 1+(−1.22−0.707i)T+(0.5+0.866i)T2 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2}
53 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
59 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
61 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
67 1+(1.22−0.707i)T+(0.5−0.866i)T2 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2}
71 1+(−0.707−0.707i)T+iT2 1 + (-0.707 - 0.707i)T + iT^{2}
73 1+(1.36+0.366i)T+(0.866+0.5i)T2 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2}
79 1+(0.258+0.965i)T+(−0.866+0.5i)T2 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2}
83 1+T2 1 + T^{2}
89 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
97 1−iT2 1 - iT^{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

−9.430474160928843441855637565767, −8.195738259904977565455691118793, −7.40449066746964296718659526842, −7.30993249873815904036546887635, −6.44752193628854747038491178194, −5.33073329913973904758818493171, −4.51244683588579382396960443648, −2.85299408165881130464906974631, −2.67487236743316888053704767631, −1.24004226250255500724432688634, 1.45260850218096057634228639266, 2.58161704036015341718477187376, 3.86835255858916967228926960950, 4.71823252058634777636533993823, 5.31119023176779212851773508432, 5.79717260686729156260461618542, 7.47873422636363537817427305353, 8.034416802055997668016947181909, 8.853631033827840468610142055005, 9.462594181931665626488406835976

Graph of the ZZ-function along the critical line