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Properties

Label 2-2548-91.51-c1-0-36
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(961,⋅)\chi_{2548} (961, \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

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

Graph of the ZZ-function along the critical line