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Properties

Label 2-1512-21.17-c1-0-17
Degree 22
Conductor 15121512
Sign 0.165−0.986i0.165 - 0.986i
Analytic cond. 12.073312.0733
Root an. cond. 3.474673.47467
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  + (1.99 + 3.46i)5-s + (0.605 + 2.57i)7-s + (4.23 + 2.44i)11-s + 0.678i·13-s + (3.57 − 6.19i)17-s + (4.19 − 2.42i)19-s + (0.556 − 0.321i)23-s + (−5.49 + 9.52i)25-s − 4.89i·29-s + (4.60 + 2.65i)31-s + (−7.70 + 7.24i)35-s + (−0.0905 − 0.156i)37-s − 9.52·41-s − 2.28·43-s + (1.62 + 2.81i)47-s + ⋯
L(s)  = 1  + (0.894 + 1.54i)5-s + (0.229 + 0.973i)7-s + (1.27 + 0.737i)11-s + 0.188i·13-s + (0.867 − 1.50i)17-s + (0.962 − 0.555i)19-s + (0.116 − 0.0670i)23-s + (−1.09 + 1.90i)25-s − 0.908i·29-s + (0.827 + 0.477i)31-s + (−1.30 + 1.22i)35-s + (−0.0148 − 0.0257i)37-s − 1.48·41-s − 0.348·43-s + (0.237 + 0.410i)47-s + ⋯

Functional equation

Λ(s)=(1512s/2 ΓC(s) L(s)=((0.165−0.986i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1512s/2 ΓC(s+1/2) L(s)=((0.165−0.986i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15121512    =    23⋅33⋅72^{3} \cdot 3^{3} \cdot 7
Sign: 0.165−0.986i0.165 - 0.986i
Analytic conductor: 12.073312.0733
Root analytic conductor: 3.474673.47467
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1512(1025,⋅)\chi_{1512} (1025, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1512, ( :1/2), 0.165−0.986i)(2,\ 1512,\ (\ :1/2),\ 0.165 - 0.986i)

Particular Values

L(1)L(1) ≈\approx 2.3231822652.323182265
L(12)L(\frac12) ≈\approx 2.3231822652.323182265
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
3 1 1
7 1+(−0.605−2.57i)T 1 + (-0.605 - 2.57i)T
good5 1+(−1.99−3.46i)T+(−2.5+4.33i)T2 1 + (-1.99 - 3.46i)T + (-2.5 + 4.33i)T^{2}
11 1+(−4.23−2.44i)T+(5.5+9.52i)T2 1 + (-4.23 - 2.44i)T + (5.5 + 9.52i)T^{2}
13 1−0.678iT−13T2 1 - 0.678iT - 13T^{2}
17 1+(−3.57+6.19i)T+(−8.5−14.7i)T2 1 + (-3.57 + 6.19i)T + (-8.5 - 14.7i)T^{2}
19 1+(−4.19+2.42i)T+(9.5−16.4i)T2 1 + (-4.19 + 2.42i)T + (9.5 - 16.4i)T^{2}
23 1+(−0.556+0.321i)T+(11.5−19.9i)T2 1 + (-0.556 + 0.321i)T + (11.5 - 19.9i)T^{2}
29 1+4.89iT−29T2 1 + 4.89iT - 29T^{2}
31 1+(−4.60−2.65i)T+(15.5+26.8i)T2 1 + (-4.60 - 2.65i)T + (15.5 + 26.8i)T^{2}
37 1+(0.0905+0.156i)T+(−18.5+32.0i)T2 1 + (0.0905 + 0.156i)T + (-18.5 + 32.0i)T^{2}
41 1+9.52T+41T2 1 + 9.52T + 41T^{2}
43 1+2.28T+43T2 1 + 2.28T + 43T^{2}
47 1+(−1.62−2.81i)T+(−23.5+40.7i)T2 1 + (-1.62 - 2.81i)T + (-23.5 + 40.7i)T^{2}
53 1+(5.96+3.44i)T+(26.5+45.8i)T2 1 + (5.96 + 3.44i)T + (26.5 + 45.8i)T^{2}
59 1+(−4.94+8.57i)T+(−29.5−51.0i)T2 1 + (-4.94 + 8.57i)T + (-29.5 - 51.0i)T^{2}
61 1+(9.15−5.28i)T+(30.5−52.8i)T2 1 + (9.15 - 5.28i)T + (30.5 - 52.8i)T^{2}
67 1+(−5.39+9.34i)T+(−33.5−58.0i)T2 1 + (-5.39 + 9.34i)T + (-33.5 - 58.0i)T^{2}
71 1+1.03iT−71T2 1 + 1.03iT - 71T^{2}
73 1+(−0.409−0.236i)T+(36.5+63.2i)T2 1 + (-0.409 - 0.236i)T + (36.5 + 63.2i)T^{2}
79 1+(−4.08−7.07i)T+(−39.5+68.4i)T2 1 + (-4.08 - 7.07i)T + (-39.5 + 68.4i)T^{2}
83 1+2.53T+83T2 1 + 2.53T + 83T^{2}
89 1+(8.90+15.4i)T+(−44.5+77.0i)T2 1 + (8.90 + 15.4i)T + (-44.5 + 77.0i)T^{2}
97 1+3.07iT−97T2 1 + 3.07iT - 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

−9.588067333553366023283870096681, −9.238060499937250790846399848462, −7.936597080635709448876989149979, −6.95050132811326981361080815336, −6.59354120249588859849883838716, −5.62784133255175442225430474287, −4.80419503809866343358086195039, −3.32021829044458824106536033005, −2.65718254880629312543134622228, −1.62751909990298823692179484041, 1.11922646058009259662409036304, 1.49108124269569695313664048530, 3.45503764217551215318687944493, 4.21207331426679290873796321340, 5.22456391485467806373520479121, 5.88795897454157092214382469734, 6.73590492794283692069152251944, 7.994797139218189136702724229135, 8.463940229441262867523709349540, 9.313574991961802970512834750744

Graph of the ZZ-function along the critical line