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Properties

Label 2-57-19.7-c1-0-0
Degree 22
Conductor 5757
Sign 0.254−0.967i0.254 - 0.967i
Analytic cond. 0.4551470.455147
Root an. cond. 0.6746460.674646
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  + (1.25 + 2.17i)2-s + (−0.5 − 0.866i)3-s + (−2.16 + 3.74i)4-s + (−1.66 − 2.87i)5-s + (1.25 − 2.17i)6-s + 2.32·7-s − 5.83·8-s + (−0.499 + 0.866i)9-s + (4.17 − 7.23i)10-s − 1.70·11-s + 4.32·12-s + (−2.01 + 3.48i)13-s + (2.91 + 5.05i)14-s + (−1.66 + 2.87i)15-s + (−3.01 − 5.22i)16-s + ⋯
L(s)  = 1  + (0.888 + 1.53i)2-s + (−0.288 − 0.499i)3-s + (−1.08 + 1.87i)4-s + (−0.742 − 1.28i)5-s + (0.513 − 0.888i)6-s + 0.877·7-s − 2.06·8-s + (−0.166 + 0.288i)9-s + (1.32 − 2.28i)10-s − 0.514·11-s + 1.24·12-s + (−0.558 + 0.967i)13-s + (0.779 + 1.35i)14-s + (−0.428 + 0.742i)15-s + (−0.753 − 1.30i)16-s + ⋯

Functional equation

Λ(s)=(57s/2 ΓC(s) L(s)=((0.254−0.967i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(57s/2 ΓC(s+1/2) L(s)=((0.254−0.967i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5757    =    3⋅193 \cdot 19
Sign: 0.254−0.967i0.254 - 0.967i
Analytic conductor: 0.4551470.455147
Root analytic conductor: 0.6746460.674646
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ57(7,⋅)\chi_{57} (7, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 57, ( :1/2), 0.254−0.967i)(2,\ 57,\ (\ :1/2),\ 0.254 - 0.967i)

Particular Values

L(1)L(1) ≈\approx 0.846487+0.652629i0.846487 + 0.652629i
L(12)L(\frac12) ≈\approx 0.846487+0.652629i0.846487 + 0.652629i
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)
bad3 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
19 1+(−0.193+4.35i)T 1 + (-0.193 + 4.35i)T
good2 1+(−1.25−2.17i)T+(−1+1.73i)T2 1 + (-1.25 - 2.17i)T + (-1 + 1.73i)T^{2}
5 1+(1.66+2.87i)T+(−2.5+4.33i)T2 1 + (1.66 + 2.87i)T + (-2.5 + 4.33i)T^{2}
7 1−2.32T+7T2 1 - 2.32T + 7T^{2}
11 1+1.70T+11T2 1 + 1.70T + 11T^{2}
13 1+(2.01−3.48i)T+(−6.5−11.2i)T2 1 + (2.01 - 3.48i)T + (-6.5 - 11.2i)T^{2}
17 1+(−8.5+14.7i)T2 1 + (-8.5 + 14.7i)T^{2}
23 1+(−1.17+2.03i)T+(−11.5−19.9i)T2 1 + (-1.17 + 2.03i)T + (-11.5 - 19.9i)T^{2}
29 1+(3.32−5.75i)T+(−14.5−25.1i)T2 1 + (3.32 - 5.75i)T + (-14.5 - 25.1i)T^{2}
31 1−6.70T+31T2 1 - 6.70T + 31T^{2}
37 1+T+37T2 1 + T + 37T^{2}
41 1+(−3.32−5.75i)T+(−20.5+35.5i)T2 1 + (-3.32 - 5.75i)T + (-20.5 + 35.5i)T^{2}
43 1+(0.353+0.612i)T+(−21.5+37.2i)T2 1 + (0.353 + 0.612i)T + (-21.5 + 37.2i)T^{2}
47 1+(−3+5.19i)T+(−23.5−40.7i)T2 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2}
53 1+(−4.98+8.62i)T+(−26.5−45.8i)T2 1 + (-4.98 + 8.62i)T + (-26.5 - 45.8i)T^{2}
59 1+(0.853+1.47i)T+(−29.5+51.0i)T2 1 + (0.853 + 1.47i)T + (-29.5 + 51.0i)T^{2}
61 1+(1.69−2.93i)T+(−30.5−52.8i)T2 1 + (1.69 - 2.93i)T + (-30.5 - 52.8i)T^{2}
67 1+(−4.18+7.25i)T+(−33.5−58.0i)T2 1 + (-4.18 + 7.25i)T + (-33.5 - 58.0i)T^{2}
71 1+(−4.70−8.15i)T+(−35.5+61.4i)T2 1 + (-4.70 - 8.15i)T + (-35.5 + 61.4i)T^{2}
73 1+(5.82+10.0i)T+(−36.5+63.2i)T2 1 + (5.82 + 10.0i)T + (-36.5 + 63.2i)T^{2}
79 1+(−1.67−2.90i)T+(−39.5+68.4i)T2 1 + (-1.67 - 2.90i)T + (-39.5 + 68.4i)T^{2}
83 1+10.0T+83T2 1 + 10.0T + 83T^{2}
89 1+(−1.33+2.32i)T+(−44.5−77.0i)T2 1 + (-1.33 + 2.32i)T + (-44.5 - 77.0i)T^{2}
97 1+(8.86+15.3i)T+(−48.5+84.0i)T2 1 + (8.86 + 15.3i)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

−15.51868418320865219810418344004, −14.46507060263240882713199873838, −13.35633797664994695896872278126, −12.49214142205402292838886570007, −11.51545091123422372035104233252, −8.818723326978585894456721069922, −7.957880378968913562745530172828, −6.90563790575456069500084841087, −5.17781613996977954792779011068, −4.51778149694231630428310060811, 2.81871623103593173270887540885, 4.14833953162062799113937923518, 5.57222427665717340383168612318, 7.77450293761169913236384536454, 9.981722622473345269012849564950, 10.74316328484034785040043227968, 11.47038008537841220191263069048, 12.39486437786918420430776271376, 13.87997974762801974695047236037, 14.81197389192486419983691683028

Graph of the ZZ-function along the critical line