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Properties

Label 2-2548-91.9-c1-0-29
Degree 22
Conductor 25482548
Sign 0.362+0.932i0.362 + 0.932i
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  + 0.421·3-s + (0.466 − 0.808i)5-s − 2.82·9-s + 0.959·11-s + (3.50 − 0.864i)13-s + (0.196 − 0.340i)15-s + (1.43 − 2.47i)17-s + 2.32·19-s + (−3.51 − 6.08i)23-s + (2.06 + 3.57i)25-s − 2.45·27-s + (−1.84 + 3.18i)29-s + (−0.286 − 0.496i)31-s + 0.404·33-s + (0.734 + 1.27i)37-s + ⋯
L(s)  = 1  + 0.243·3-s + (0.208 − 0.361i)5-s − 0.940·9-s + 0.289·11-s + (0.970 − 0.239i)13-s + (0.0507 − 0.0879i)15-s + (0.346 − 0.600i)17-s + 0.533·19-s + (−0.732 − 1.26i)23-s + (0.412 + 0.715i)25-s − 0.472·27-s + (−0.341 + 0.592i)29-s + (−0.0514 − 0.0891i)31-s + 0.0703·33-s + (0.120 + 0.209i)37-s + ⋯

Functional equation

Λ(s)=(2548s/2 ΓC(s) L(s)=((0.362+0.932i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.362 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2548s/2 ΓC(s+1/2) L(s)=((0.362+0.932i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.362 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25482548    =    22⋅72⋅132^{2} \cdot 7^{2} \cdot 13
Sign: 0.362+0.932i0.362 + 0.932i
Analytic conductor: 20.345820.3458
Root analytic conductor: 4.510644.51064
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2548(373,⋅)\chi_{2548} (373, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2548, ( :1/2), 0.362+0.932i)(2,\ 2548,\ (\ :1/2),\ 0.362 + 0.932i)

Particular Values

L(1)L(1) ≈\approx 1.8250065191.825006519
L(12)L(\frac12) ≈\approx 1.8250065191.825006519
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+(−3.50+0.864i)T 1 + (-3.50 + 0.864i)T
good3 1−0.421T+3T2 1 - 0.421T + 3T^{2}
5 1+(−0.466+0.808i)T+(−2.5−4.33i)T2 1 + (-0.466 + 0.808i)T + (-2.5 - 4.33i)T^{2}
11 1−0.959T+11T2 1 - 0.959T + 11T^{2}
17 1+(−1.43+2.47i)T+(−8.5−14.7i)T2 1 + (-1.43 + 2.47i)T + (-8.5 - 14.7i)T^{2}
19 1−2.32T+19T2 1 - 2.32T + 19T^{2}
23 1+(3.51+6.08i)T+(−11.5+19.9i)T2 1 + (3.51 + 6.08i)T + (-11.5 + 19.9i)T^{2}
29 1+(1.84−3.18i)T+(−14.5−25.1i)T2 1 + (1.84 - 3.18i)T + (-14.5 - 25.1i)T^{2}
31 1+(0.286+0.496i)T+(−15.5+26.8i)T2 1 + (0.286 + 0.496i)T + (-15.5 + 26.8i)T^{2}
37 1+(−0.734−1.27i)T+(−18.5+32.0i)T2 1 + (-0.734 - 1.27i)T + (-18.5 + 32.0i)T^{2}
41 1+(−5.42+9.39i)T+(−20.5−35.5i)T2 1 + (-5.42 + 9.39i)T + (-20.5 - 35.5i)T^{2}
43 1+(1.63+2.82i)T+(−21.5+37.2i)T2 1 + (1.63 + 2.82i)T + (-21.5 + 37.2i)T^{2}
47 1+(1.23−2.13i)T+(−23.5−40.7i)T2 1 + (1.23 - 2.13i)T + (-23.5 - 40.7i)T^{2}
53 1+(3.81+6.59i)T+(−26.5+45.8i)T2 1 + (3.81 + 6.59i)T + (-26.5 + 45.8i)T^{2}
59 1+(−3.11+5.39i)T+(−29.5−51.0i)T2 1 + (-3.11 + 5.39i)T + (-29.5 - 51.0i)T^{2}
61 1+7.96T+61T2 1 + 7.96T + 61T^{2}
67 1−11.5T+67T2 1 - 11.5T + 67T^{2}
71 1+(−1.23−2.14i)T+(−35.5+61.4i)T2 1 + (-1.23 - 2.14i)T + (-35.5 + 61.4i)T^{2}
73 1+(−3.49−6.04i)T+(−36.5+63.2i)T2 1 + (-3.49 - 6.04i)T + (-36.5 + 63.2i)T^{2}
79 1+(−7.93+13.7i)T+(−39.5−68.4i)T2 1 + (-7.93 + 13.7i)T + (-39.5 - 68.4i)T^{2}
83 1−5.06T+83T2 1 - 5.06T + 83T^{2}
89 1+(8.07+13.9i)T+(−44.5+77.0i)T2 1 + (8.07 + 13.9i)T + (-44.5 + 77.0i)T^{2}
97 1+(4.24+7.35i)T+(−48.5+84.0i)T2 1 + (4.24 + 7.35i)T + (-48.5 + 84.0i)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

−8.766938895284323522135440502906, −8.163345210190629533430124000852, −7.29106315974764795660466329138, −6.34387002205453269490338279203, −5.63650960521565248025164527934, −4.92937244810118772264922204621, −3.76938808730666709464803914721, −3.05348677777688990068277012770, −1.93108740442817451232396623356, −0.63314550436035999645068591475, 1.23718970577902190617976156284, 2.41266971783672096046632228079, 3.36457701842827924374720147000, 4.04903215829355983589076611543, 5.28915167537721041187916603987, 6.06545023077443941437559878578, 6.53030355114039450480176757102, 7.80344070582286772518771918085, 8.125050838358834213754152178073, 9.142246132614663045995523349847

Graph of the ZZ-function along the critical line