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Properties

Label 2-2548-91.30-c1-0-17
Degree 22
Conductor 25482548
Sign 0.643+0.765i0.643 + 0.765i
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  − 0.605·3-s + (−3.48 − 2.01i)5-s − 2.63·9-s + 3.01i·11-s + (1.03 + 3.45i)13-s + (2.11 + 1.21i)15-s + (2.53 − 4.38i)17-s + 3.31i·19-s + (−2.45 − 4.25i)23-s + (5.60 + 9.71i)25-s + 3.41·27-s + (−1.30 + 2.26i)29-s + (−8.86 + 5.11i)31-s − 1.82i·33-s + (−1.76 + 1.02i)37-s + ⋯
L(s)  = 1  − 0.349·3-s + (−1.55 − 0.900i)5-s − 0.877·9-s + 0.909i·11-s + (0.285 + 0.958i)13-s + (0.545 + 0.314i)15-s + (0.614 − 1.06i)17-s + 0.761i·19-s + (−0.511 − 0.886i)23-s + (1.12 + 1.94i)25-s + 0.656·27-s + (−0.243 + 0.421i)29-s + (−1.59 + 0.919i)31-s − 0.318i·33-s + (−0.290 + 0.167i)37-s + ⋯

Functional equation

Λ(s)=(2548s/2 ΓC(s) L(s)=((0.643+0.765i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 + 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2548s/2 ΓC(s+1/2) L(s)=((0.643+0.765i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.643 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25482548    =    22⋅72⋅132^{2} \cdot 7^{2} \cdot 13
Sign: 0.643+0.765i0.643 + 0.765i
Analytic conductor: 20.345820.3458
Root analytic conductor: 4.510644.51064
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2548(1941,⋅)\chi_{2548} (1941, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2548, ( :1/2), 0.643+0.765i)(2,\ 2548,\ (\ :1/2),\ 0.643 + 0.765i)

Particular Values

L(1)L(1) ≈\approx 0.67108467650.6710846765
L(12)L(\frac12) ≈\approx 0.67108467650.6710846765
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+(−1.03−3.45i)T 1 + (-1.03 - 3.45i)T
good3 1+0.605T+3T2 1 + 0.605T + 3T^{2}
5 1+(3.48+2.01i)T+(2.5+4.33i)T2 1 + (3.48 + 2.01i)T + (2.5 + 4.33i)T^{2}
11 1−3.01iT−11T2 1 - 3.01iT - 11T^{2}
17 1+(−2.53+4.38i)T+(−8.5−14.7i)T2 1 + (-2.53 + 4.38i)T + (-8.5 - 14.7i)T^{2}
19 1−3.31iT−19T2 1 - 3.31iT - 19T^{2}
23 1+(2.45+4.25i)T+(−11.5+19.9i)T2 1 + (2.45 + 4.25i)T + (-11.5 + 19.9i)T^{2}
29 1+(1.30−2.26i)T+(−14.5−25.1i)T2 1 + (1.30 - 2.26i)T + (-14.5 - 25.1i)T^{2}
31 1+(8.86−5.11i)T+(15.5−26.8i)T2 1 + (8.86 - 5.11i)T + (15.5 - 26.8i)T^{2}
37 1+(1.76−1.02i)T+(18.5−32.0i)T2 1 + (1.76 - 1.02i)T + (18.5 - 32.0i)T^{2}
41 1+(0.252+0.145i)T+(20.5+35.5i)T2 1 + (0.252 + 0.145i)T + (20.5 + 35.5i)T^{2}
43 1+(0.581+1.00i)T+(−21.5+37.2i)T2 1 + (0.581 + 1.00i)T + (-21.5 + 37.2i)T^{2}
47 1+(3.64+2.10i)T+(23.5+40.7i)T2 1 + (3.64 + 2.10i)T + (23.5 + 40.7i)T^{2}
53 1+(1.74+3.02i)T+(−26.5+45.8i)T2 1 + (1.74 + 3.02i)T + (-26.5 + 45.8i)T^{2}
59 1+(−5.84−3.37i)T+(29.5+51.0i)T2 1 + (-5.84 - 3.37i)T + (29.5 + 51.0i)T^{2}
61 1−13.2T+61T2 1 - 13.2T + 61T^{2}
67 1+4.13iT−67T2 1 + 4.13iT - 67T^{2}
71 1+(1.10−0.639i)T+(35.5−61.4i)T2 1 + (1.10 - 0.639i)T + (35.5 - 61.4i)T^{2}
73 1+(−12.9+7.45i)T+(36.5−63.2i)T2 1 + (-12.9 + 7.45i)T + (36.5 - 63.2i)T^{2}
79 1+(−3.45+5.99i)T+(−39.5−68.4i)T2 1 + (-3.45 + 5.99i)T + (-39.5 - 68.4i)T^{2}
83 1+10.4iT−83T2 1 + 10.4iT - 83T^{2}
89 1+(0.511−0.295i)T+(44.5−77.0i)T2 1 + (0.511 - 0.295i)T + (44.5 - 77.0i)T^{2}
97 1+(−4.94+2.85i)T+(48.5−84.0i)T2 1 + (-4.94 + 2.85i)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

−8.722639610079831158155775050273, −8.082009713110955019091504938372, −7.31951955211937855641090244151, −6.65949500311770359550624017973, −5.40224146704504512395703896402, −4.88652456031013094910494971603, −4.02039036807863898897089941785, −3.29791698587464063649556136938, −1.80533337343142112157151878977, −0.41104898762986041707962024922, 0.64276366222519664525810203679, 2.55889817464157630651148417071, 3.59568360903184840803012659573, 3.75096454430883935203744346952, 5.27078029917858992325624176618, 5.88985137049404324602837030232, 6.71029666761524988296470659311, 7.67841815089566583086028485781, 8.084326513952875366014526399660, 8.723074717159982645813843794924

Graph of the ZZ-function along the critical line