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Properties

Label 2-507-39.5-c1-0-14
Degree 22
Conductor 507507
Sign −0.0125−0.999i-0.0125 - 0.999i
Analytic cond. 4.048414.04841
Root an. cond. 2.012062.01206
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  + (0.928 + 0.928i)2-s + (−1.37 + 1.05i)3-s − 0.276i·4-s + (2.12 + 2.12i)5-s + (−2.25 − 0.293i)6-s + (2.06 + 2.06i)7-s + (2.11 − 2.11i)8-s + (0.767 − 2.90i)9-s + 3.94i·10-s + (1.88 − 1.88i)11-s + (0.291 + 0.379i)12-s + 3.83i·14-s + (−5.16 − 0.671i)15-s + 3.37·16-s + 0.198·17-s + (3.40 − 1.98i)18-s + ⋯
L(s)  = 1  + (0.656 + 0.656i)2-s + (−0.792 + 0.610i)3-s − 0.138i·4-s + (0.950 + 0.950i)5-s + (−0.920 − 0.119i)6-s + (0.780 + 0.780i)7-s + (0.747 − 0.747i)8-s + (0.255 − 0.966i)9-s + 1.24i·10-s + (0.568 − 0.568i)11-s + (0.0842 + 0.109i)12-s + 1.02i·14-s + (−1.33 − 0.173i)15-s + 0.842·16-s + 0.0482·17-s + (0.802 − 0.466i)18-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((−0.0125−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0125 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((−0.0125−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0125 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: −0.0125−0.999i-0.0125 - 0.999i
Analytic conductor: 4.048414.04841
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ507(239,⋅)\chi_{507} (239, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 507, ( :1/2), −0.0125−0.999i)(2,\ 507,\ (\ :1/2),\ -0.0125 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 1.43186+1.44994i1.43186 + 1.44994i
L(12)L(\frac12) ≈\approx 1.43186+1.44994i1.43186 + 1.44994i
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+(1.37−1.05i)T 1 + (1.37 - 1.05i)T
13 1 1
good2 1+(−0.928−0.928i)T+2iT2 1 + (-0.928 - 0.928i)T + 2iT^{2}
5 1+(−2.12−2.12i)T+5iT2 1 + (-2.12 - 2.12i)T + 5iT^{2}
7 1+(−2.06−2.06i)T+7iT2 1 + (-2.06 - 2.06i)T + 7iT^{2}
11 1+(−1.88+1.88i)T−11iT2 1 + (-1.88 + 1.88i)T - 11iT^{2}
17 1−0.198T+17T2 1 - 0.198T + 17T^{2}
19 1+(3.75−3.75i)T−19iT2 1 + (3.75 - 3.75i)T - 19iT^{2}
23 1+3.30T+23T2 1 + 3.30T + 23T^{2}
29 1−3.73iT−29T2 1 - 3.73iT - 29T^{2}
31 1+(−0.550+0.550i)T−31iT2 1 + (-0.550 + 0.550i)T - 31iT^{2}
37 1+(3.60+3.60i)T+37iT2 1 + (3.60 + 3.60i)T + 37iT^{2}
41 1+(2.69+2.69i)T+41iT2 1 + (2.69 + 2.69i)T + 41iT^{2}
43 1−11.6iT−43T2 1 - 11.6iT - 43T^{2}
47 1+(−4.29+4.29i)T−47iT2 1 + (-4.29 + 4.29i)T - 47iT^{2}
53 1+13.6iT−53T2 1 + 13.6iT - 53T^{2}
59 1+(2.36−2.36i)T−59iT2 1 + (2.36 - 2.36i)T - 59iT^{2}
61 1+2.70T+61T2 1 + 2.70T + 61T^{2}
67 1+(−5.33+5.33i)T−67iT2 1 + (-5.33 + 5.33i)T - 67iT^{2}
71 1+(3.78+3.78i)T+71iT2 1 + (3.78 + 3.78i)T + 71iT^{2}
73 1+(−3.46−3.46i)T+73iT2 1 + (-3.46 - 3.46i)T + 73iT^{2}
79 1+11.1T+79T2 1 + 11.1T + 79T^{2}
83 1+(1.84+1.84i)T+83iT2 1 + (1.84 + 1.84i)T + 83iT^{2}
89 1+(0.776−0.776i)T−89iT2 1 + (0.776 - 0.776i)T - 89iT^{2}
97 1+(−8.60+8.60i)T−97iT2 1 + (-8.60 + 8.60i)T - 97iT^{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

−11.00036140137645342390140100966, −10.34662819234942939477266290546, −9.619545733537231157799732012086, −8.470319233434678909781224306203, −6.98088903781774988601365790201, −6.13399640481942184408801124178, −5.76202590096403062691811913157, −4.80536607738604064787716223104, −3.62109483445524268991347905491, −1.81407869116434658819789541082, 1.32767855862334024971008426756, 2.20940547336661745047875790972, 4.25730382377750002943264211063, 4.78052172749322086259042023364, 5.77319312371933257610586110926, 6.96220784112052689657845979392, 7.902231804409031009363809678894, 8.885690847205571661293027743411, 10.18018039723633150654700129585, 10.88261854811920375819331984979

Graph of the ZZ-function along the critical line