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Properties

Label 2-507-13.3-c1-0-2
Degree 22
Conductor 507507
Sign 0.522+0.852i0.522 + 0.852i
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  + (−1.20 − 2.09i)2-s + (−0.5 − 0.866i)3-s + (−1.91 + 3.31i)4-s − 2.82·5-s + (−1.20 + 2.09i)6-s + (−1.41 + 2.44i)7-s + 4.41·8-s + (−0.499 + 0.866i)9-s + (3.41 + 5.91i)10-s + (−1 − 1.73i)11-s + 3.82·12-s + 6.82·14-s + (1.41 + 2.44i)15-s + (−1.49 − 2.59i)16-s + (1.82 − 3.16i)17-s + 2.41·18-s + ⋯
L(s)  = 1  + (−0.853 − 1.47i)2-s + (−0.288 − 0.499i)3-s + (−0.957 + 1.65i)4-s − 1.26·5-s + (−0.492 + 0.853i)6-s + (−0.534 + 0.925i)7-s + 1.56·8-s + (−0.166 + 0.288i)9-s + (1.07 + 1.87i)10-s + (−0.301 − 0.522i)11-s + 1.10·12-s + 1.82·14-s + (0.365 + 0.632i)15-s + (−0.374 − 0.649i)16-s + (0.443 − 0.768i)17-s + 0.569·18-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((0.522+0.852i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+1/2) L(s)=((0.522+0.852i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: 0.522+0.852i0.522 + 0.852i
Analytic conductor: 4.048414.04841
Root analytic conductor: 2.012062.01206
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ507(484,⋅)\chi_{507} (484, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 507, ( :1/2), 0.522+0.852i)(2,\ 507,\ (\ :1/2),\ 0.522 + 0.852i)

Particular Values

L(1)L(1) ≈\approx 0.368101−0.206275i0.368101 - 0.206275i
L(12)L(\frac12) ≈\approx 0.368101−0.206275i0.368101 - 0.206275i
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
13 1 1
good2 1+(1.20+2.09i)T+(−1+1.73i)T2 1 + (1.20 + 2.09i)T + (-1 + 1.73i)T^{2}
5 1+2.82T+5T2 1 + 2.82T + 5T^{2}
7 1+(1.41−2.44i)T+(−3.5−6.06i)T2 1 + (1.41 - 2.44i)T + (-3.5 - 6.06i)T^{2}
11 1+(1+1.73i)T+(−5.5+9.52i)T2 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2}
17 1+(−1.82+3.16i)T+(−8.5−14.7i)T2 1 + (-1.82 + 3.16i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1.41+2.44i)T+(−9.5−16.4i)T2 1 + (-1.41 + 2.44i)T + (-9.5 - 16.4i)T^{2}
23 1+(−2−3.46i)T+(−11.5+19.9i)T2 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2}
29 1+(1+1.73i)T+(−14.5+25.1i)T2 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2}
31 1−6.82T+31T2 1 - 6.82T + 31T^{2}
37 1+(−1.82−3.16i)T+(−18.5+32.0i)T2 1 + (-1.82 - 3.16i)T + (-18.5 + 32.0i)T^{2}
41 1+(−5.41−9.37i)T+(−20.5+35.5i)T2 1 + (-5.41 - 9.37i)T + (-20.5 + 35.5i)T^{2}
43 1+(4.82−8.36i)T+(−21.5−37.2i)T2 1 + (4.82 - 8.36i)T + (-21.5 - 37.2i)T^{2}
47 1−0.343T+47T2 1 - 0.343T + 47T^{2}
53 1+2T+53T2 1 + 2T + 53T^{2}
59 1+(1.82−3.16i)T+(−29.5−51.0i)T2 1 + (1.82 - 3.16i)T + (-29.5 - 51.0i)T^{2}
61 1+(−4.65+8.06i)T+(−30.5−52.8i)T2 1 + (-4.65 + 8.06i)T + (-30.5 - 52.8i)T^{2}
67 1+(−0.585−1.01i)T+(−33.5+58.0i)T2 1 + (-0.585 - 1.01i)T + (-33.5 + 58.0i)T^{2}
71 1+(−1+1.73i)T+(−35.5−61.4i)T2 1 + (-1 + 1.73i)T + (-35.5 - 61.4i)T^{2}
73 1+11.6T+73T2 1 + 11.6T + 73T^{2}
79 1−11.3T+79T2 1 - 11.3T + 79T^{2}
83 1−7.65T+83T2 1 - 7.65T + 83T^{2}
89 1+(−4.58−7.94i)T+(−44.5+77.0i)T2 1 + (-4.58 - 7.94i)T + (-44.5 + 77.0i)T^{2}
97 1+(3.82−6.63i)T+(−48.5−84.0i)T2 1 + (3.82 - 6.63i)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

−11.13222969888195980429462597313, −9.883618923950290099840161393058, −9.173615530531791896315117298857, −8.189742476520684086030066841045, −7.63766527649877479335388198009, −6.24109139264742190503335946828, −4.79152638643166828516196947337, −3.33327621237102495705815664279, −2.68088528151026880490290176226, −0.878569479498000340097638638849, 0.51156222768782491682817702019, 3.61134353718673147896807893578, 4.53521804362827743023511464805, 5.69883922840257998416938983799, 6.78441309465247104042041703103, 7.44887170621006871566221752535, 8.158344428701247027218031688698, 9.053173692677022045589583625512, 10.17901611257579890689381422858, 10.51147240422101642633235870186

Graph of the ZZ-function along the critical line