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Properties

Label 2-48-48.35-c1-0-0
Degree 22
Conductor 4848
Sign 0.627−0.778i0.627 - 0.778i
Analytic cond. 0.3832810.383281
Root an. cond. 0.6190970.619097
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  + (−1.39 − 0.204i)2-s + (−0.814 + 1.52i)3-s + (1.91 + 0.573i)4-s + (2.08 + 2.08i)5-s + (1.45 − 1.97i)6-s − 1.14·7-s + (−2.56 − 1.19i)8-s + (−1.67 − 2.48i)9-s + (−2.48 − 3.34i)10-s + (1.67 − 1.67i)11-s + (−2.43 + 2.46i)12-s + (0.146 + 0.146i)13-s + (1.60 + 0.234i)14-s + (−4.88 + 1.48i)15-s + (3.34 + 2.19i)16-s − 5.59i·17-s + ⋯
L(s)  = 1  + (−0.989 − 0.144i)2-s + (−0.470 + 0.882i)3-s + (0.958 + 0.286i)4-s + (0.931 + 0.931i)5-s + (0.592 − 0.805i)6-s − 0.433·7-s + (−0.906 − 0.422i)8-s + (−0.558 − 0.829i)9-s + (−0.787 − 1.05i)10-s + (0.504 − 0.504i)11-s + (−0.703 + 0.710i)12-s + (0.0405 + 0.0405i)13-s + (0.428 + 0.0627i)14-s + (−1.26 + 0.384i)15-s + (0.835 + 0.549i)16-s − 1.35i·17-s + ⋯

Functional equation

Λ(s)=(48s/2 ΓC(s) L(s)=((0.627−0.778i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(48s/2 ΓC(s+1/2) L(s)=((0.627−0.778i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4848    =    24⋅32^{4} \cdot 3
Sign: 0.627−0.778i0.627 - 0.778i
Analytic conductor: 0.3832810.383281
Root analytic conductor: 0.6190970.619097
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ48(35,⋅)\chi_{48} (35, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 48, ( :1/2), 0.627−0.778i)(2,\ 48,\ (\ :1/2),\ 0.627 - 0.778i)

Particular Values

L(1)L(1) ≈\approx 0.489378+0.234233i0.489378 + 0.234233i
L(12)L(\frac12) ≈\approx 0.489378+0.234233i0.489378 + 0.234233i
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+(1.39+0.204i)T 1 + (1.39 + 0.204i)T
3 1+(0.814−1.52i)T 1 + (0.814 - 1.52i)T
good5 1+(−2.08−2.08i)T+5iT2 1 + (-2.08 - 2.08i)T + 5iT^{2}
7 1+1.14T+7T2 1 + 1.14T + 7T^{2}
11 1+(−1.67+1.67i)T−11iT2 1 + (-1.67 + 1.67i)T - 11iT^{2}
13 1+(−0.146−0.146i)T+13iT2 1 + (-0.146 - 0.146i)T + 13iT^{2}
17 1+5.59iT−17T2 1 + 5.59iT - 17T^{2}
19 1+(−1.48+1.48i)T−19iT2 1 + (-1.48 + 1.48i)T - 19iT^{2}
23 1−3.34iT−23T2 1 - 3.34iT - 23T^{2}
29 1+(3.51−3.51i)T−29iT2 1 + (3.51 - 3.51i)T - 29iT^{2}
31 1+5.83iT−31T2 1 + 5.83iT - 31T^{2}
37 1+(4.83−4.83i)T−37iT2 1 + (4.83 - 4.83i)T - 37iT^{2}
41 1−0.610T+41T2 1 - 0.610T + 41T^{2}
43 1+(1.48+1.48i)T+43iT2 1 + (1.48 + 1.48i)T + 43iT^{2}
47 1−6.41T+47T2 1 - 6.41T + 47T^{2}
53 1+(−0.164−0.164i)T+53iT2 1 + (-0.164 - 0.164i)T + 53iT^{2}
59 1+(9.05−9.05i)T−59iT2 1 + (9.05 - 9.05i)T - 59iT^{2}
61 1+(−4.53−4.53i)T+61iT2 1 + (-4.53 - 4.53i)T + 61iT^{2}
67 1+(0.635−0.635i)T−67iT2 1 + (0.635 - 0.635i)T - 67iT^{2}
71 1+6.90iT−71T2 1 + 6.90iT - 71T^{2}
73 1+7.07iT−73T2 1 + 7.07iT - 73T^{2}
79 1−9.83iT−79T2 1 - 9.83iT - 79T^{2}
83 1+(8.09+8.09i)T+83iT2 1 + (8.09 + 8.09i)T + 83iT^{2}
89 1−0.490T+89T2 1 - 0.490T + 89T^{2}
97 1−12.3T+97T2 1 - 12.3T + 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

−16.11857758571428158175444325836, −15.04570204789720383423722336662, −13.76223171002311163171837455596, −11.80782650961055644880870632247, −10.89996512231204696049612845669, −9.839330008068927701384528138533, −9.141657938539346332277417531504, −6.99389724778086933820593516998, −5.83526797377675345309218897203, −3.10481947319434072200526019304, 1.69578359853725675531896880474, 5.63088384160470586525400218939, 6.69955058569404092588748504954, 8.235920566746012977433715157663, 9.404737451782279877588433927233, 10.63361851716249816660754198814, 12.17940975584803982055138831264, 12.93268888283903553826811291185, 14.38784783441989261318263004208, 16.04152980899198313282517734703

Graph of the ZZ-function along the critical line