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Properties

Label 2-2548-91.25-c1-0-12
Degree 22
Conductor 25482548
Sign −0.265−0.964i-0.265 - 0.964i
Analytic cond. 20.345820.3458
Root an. cond. 4.510644.51064
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  + (−0.646 + 1.12i)3-s + (2.42 − 1.39i)5-s + (0.663 + 1.14i)9-s + (−0.894 − 0.516i)11-s + (1.79 + 3.12i)13-s + 3.62i·15-s + (−3.47 + 6.02i)17-s + (2.19 − 1.26i)19-s + (1.91 + 3.31i)23-s + (1.41 − 2.45i)25-s − 5.59·27-s − 5.10·29-s + (−3.28 − 1.89i)31-s + (1.15 − 0.668i)33-s + (−5.37 + 3.10i)37-s + ⋯
L(s)  = 1  + (−0.373 + 0.646i)3-s + (1.08 − 0.625i)5-s + (0.221 + 0.382i)9-s + (−0.269 − 0.155i)11-s + (0.498 + 0.866i)13-s + 0.934i·15-s + (−0.843 + 1.46i)17-s + (0.504 − 0.291i)19-s + (0.399 + 0.691i)23-s + (0.283 − 0.490i)25-s − 1.07·27-s − 0.947·29-s + (−0.589 − 0.340i)31-s + (0.201 − 0.116i)33-s + (−0.883 + 0.510i)37-s + ⋯

Functional equation

Λ(s)=(2548s/2 ΓC(s) L(s)=((−0.265−0.964i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2548s/2 ΓC(s+1/2) L(s)=((−0.265−0.964i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25482548    =    22⋅72⋅132^{2} \cdot 7^{2} \cdot 13
Sign: −0.265−0.964i-0.265 - 0.964i
Analytic conductor: 20.345820.3458
Root analytic conductor: 4.510644.51064
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2548(753,⋅)\chi_{2548} (753, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2548, ( :1/2), −0.265−0.964i)(2,\ 2548,\ (\ :1/2),\ -0.265 - 0.964i)

Particular Values

L(1)L(1) ≈\approx 1.6313459421.631345942
L(12)L(\frac12) ≈\approx 1.6313459421.631345942
L(32)L(\frac{3}{2}) 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 1
13 1+(−1.79−3.12i)T 1 + (-1.79 - 3.12i)T
good3 1+(0.646−1.12i)T+(−1.5−2.59i)T2 1 + (0.646 - 1.12i)T + (-1.5 - 2.59i)T^{2}
5 1+(−2.42+1.39i)T+(2.5−4.33i)T2 1 + (-2.42 + 1.39i)T + (2.5 - 4.33i)T^{2}
11 1+(0.894+0.516i)T+(5.5+9.52i)T2 1 + (0.894 + 0.516i)T + (5.5 + 9.52i)T^{2}
17 1+(3.47−6.02i)T+(−8.5−14.7i)T2 1 + (3.47 - 6.02i)T + (-8.5 - 14.7i)T^{2}
19 1+(−2.19+1.26i)T+(9.5−16.4i)T2 1 + (-2.19 + 1.26i)T + (9.5 - 16.4i)T^{2}
23 1+(−1.91−3.31i)T+(−11.5+19.9i)T2 1 + (-1.91 - 3.31i)T + (-11.5 + 19.9i)T^{2}
29 1+5.10T+29T2 1 + 5.10T + 29T^{2}
31 1+(3.28+1.89i)T+(15.5+26.8i)T2 1 + (3.28 + 1.89i)T + (15.5 + 26.8i)T^{2}
37 1+(5.37−3.10i)T+(18.5−32.0i)T2 1 + (5.37 - 3.10i)T + (18.5 - 32.0i)T^{2}
41 1+0.990iT−41T2 1 + 0.990iT - 41T^{2}
43 1−0.560T+43T2 1 - 0.560T + 43T^{2}
47 1+(−6.22+3.59i)T+(23.5−40.7i)T2 1 + (-6.22 + 3.59i)T + (23.5 - 40.7i)T^{2}
53 1+(0.589−1.02i)T+(−26.5−45.8i)T2 1 + (0.589 - 1.02i)T + (-26.5 - 45.8i)T^{2}
59 1+(−10.3−5.96i)T+(29.5+51.0i)T2 1 + (-10.3 - 5.96i)T + (29.5 + 51.0i)T^{2}
61 1+(−6.92−11.9i)T+(−30.5+52.8i)T2 1 + (-6.92 - 11.9i)T + (-30.5 + 52.8i)T^{2}
67 1+(11.9+6.89i)T+(33.5+58.0i)T2 1 + (11.9 + 6.89i)T + (33.5 + 58.0i)T^{2}
71 1−3.80iT−71T2 1 - 3.80iT - 71T^{2}
73 1+(−2.76−1.59i)T+(36.5+63.2i)T2 1 + (-2.76 - 1.59i)T + (36.5 + 63.2i)T^{2}
79 1+(3.66+6.35i)T+(−39.5+68.4i)T2 1 + (3.66 + 6.35i)T + (-39.5 + 68.4i)T^{2}
83 1−16.6iT−83T2 1 - 16.6iT - 83T^{2}
89 1+(−10.3+5.94i)T+(44.5−77.0i)T2 1 + (-10.3 + 5.94i)T + (44.5 - 77.0i)T^{2}
97 1−8.60iT−97T2 1 - 8.60iT - 97T^{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

−9.078285914610475076700582186403, −8.717432002620776791420848986587, −7.58122720920630288941467461571, −6.70251679431864288281456880162, −5.68782123378028480119407910179, −5.41849699199180869750751303693, −4.39591763131123108437421313211, −3.72156410900588993217823825890, −2.17956309832178880107563434532, −1.46681555373175937534649874508, 0.55285574257926996945556979252, 1.82227211736056268478490712343, 2.69515707420621254601574439080, 3.66859753443529566543017811438, 5.02142859601844271036803068104, 5.68566768357019661662208232006, 6.41268858195890665448981243316, 7.05598536639200086786370392464, 7.62945241813893774154822081920, 8.817465223685678971461965385490

Graph of the ZZ-function along the critical line