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Properties

Label 2-13e2-13.11-c2-0-6
Degree 22
Conductor 169169
Sign −0.504−0.863i-0.504 - 0.863i
Analytic cond. 4.604914.60491
Root an. cond. 2.145902.14590
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.515 + 1.92i)2-s + (0.299 + 0.519i)3-s + (0.0309 + 0.0178i)4-s + (5.65 + 5.65i)5-s + (−1.15 + 0.308i)6-s + (−0.526 − 1.96i)7-s + (−5.68 + 5.68i)8-s + (4.32 − 7.48i)9-s + (−13.7 + 7.95i)10-s + (14.0 + 3.76i)11-s + 0.0214i·12-s + 4.05·14-s + (−1.24 + 4.62i)15-s + (−7.92 − 13.7i)16-s + (−10.2 − 5.90i)17-s + (12.1 + 12.1i)18-s + ⋯
L(s)  = 1  + (−0.257 + 0.961i)2-s + (0.0999 + 0.173i)3-s + (0.00773 + 0.00446i)4-s + (1.13 + 1.13i)5-s + (−0.192 + 0.0514i)6-s + (−0.0752 − 0.280i)7-s + (−0.710 + 0.710i)8-s + (0.480 − 0.831i)9-s + (−1.37 + 0.795i)10-s + (1.27 + 0.342i)11-s + 0.00178i·12-s + 0.289·14-s + (−0.0826 + 0.308i)15-s + (−0.495 − 0.858i)16-s + (−0.601 − 0.347i)17-s + (0.675 + 0.675i)18-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=((−0.504−0.863i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+1) L(s)=((−0.504−0.863i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: −0.504−0.863i-0.504 - 0.863i
Analytic conductor: 4.604914.60491
Root analytic conductor: 2.145902.14590
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ169(89,⋅)\chi_{169} (89, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 169, ( :1), −0.504−0.863i)(2,\ 169,\ (\ :1),\ -0.504 - 0.863i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.859149+1.49704i0.859149 + 1.49704i
L(12)L(\frac12) ≈\approx 0.859149+1.49704i0.859149 + 1.49704i
L(2)L(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)
bad13 1 1
good2 1+(0.515−1.92i)T+(−3.46−2i)T2 1 + (0.515 - 1.92i)T + (-3.46 - 2i)T^{2}
3 1+(−0.299−0.519i)T+(−4.5+7.79i)T2 1 + (-0.299 - 0.519i)T + (-4.5 + 7.79i)T^{2}
5 1+(−5.65−5.65i)T+25iT2 1 + (-5.65 - 5.65i)T + 25iT^{2}
7 1+(0.526+1.96i)T+(−42.4+24.5i)T2 1 + (0.526 + 1.96i)T + (-42.4 + 24.5i)T^{2}
11 1+(−14.0−3.76i)T+(104.+60.5i)T2 1 + (-14.0 - 3.76i)T + (104. + 60.5i)T^{2}
17 1+(10.2+5.90i)T+(144.5+250.i)T2 1 + (10.2 + 5.90i)T + (144.5 + 250. i)T^{2}
19 1+(23.4−6.28i)T+(312.−180.5i)T2 1 + (23.4 - 6.28i)T + (312. - 180.5i)T^{2}
23 1+(−0.229+0.132i)T+(264.5−458.i)T2 1 + (-0.229 + 0.132i)T + (264.5 - 458. i)T^{2}
29 1+(3.60+6.25i)T+(−420.5+728.i)T2 1 + (3.60 + 6.25i)T + (-420.5 + 728. i)T^{2}
31 1+(30.2+30.2i)T+961iT2 1 + (30.2 + 30.2i)T + 961iT^{2}
37 1+(−13.1−3.53i)T+(1.18e3+684.5i)T2 1 + (-13.1 - 3.53i)T + (1.18e3 + 684.5i)T^{2}
41 1+(−9.00+33.6i)T+(−1.45e3−840.5i)T2 1 + (-9.00 + 33.6i)T + (-1.45e3 - 840.5i)T^{2}
43 1+(−35.0−20.2i)T+(924.5+1.60e3i)T2 1 + (-35.0 - 20.2i)T + (924.5 + 1.60e3i)T^{2}
47 1+(9.87−9.87i)T−2.20e3iT2 1 + (9.87 - 9.87i)T - 2.20e3iT^{2}
53 1−77.1T+2.80e3T2 1 - 77.1T + 2.80e3T^{2}
59 1+(9.32+34.8i)T+(−3.01e3+1.74e3i)T2 1 + (9.32 + 34.8i)T + (-3.01e3 + 1.74e3i)T^{2}
61 1+(−15.4+26.7i)T+(−1.86e3−3.22e3i)T2 1 + (-15.4 + 26.7i)T + (-1.86e3 - 3.22e3i)T^{2}
67 1+(16.1−60.4i)T+(−3.88e3−2.24e3i)T2 1 + (16.1 - 60.4i)T + (-3.88e3 - 2.24e3i)T^{2}
71 1+(−8.34+2.23i)T+(4.36e3−2.52e3i)T2 1 + (-8.34 + 2.23i)T + (4.36e3 - 2.52e3i)T^{2}
73 1+(−23.3+23.3i)T−5.32e3iT2 1 + (-23.3 + 23.3i)T - 5.32e3iT^{2}
79 1+49.8T+6.24e3T2 1 + 49.8T + 6.24e3T^{2}
83 1+(60.7+60.7i)T+6.88e3iT2 1 + (60.7 + 60.7i)T + 6.88e3iT^{2}
89 1+(−85.6−22.9i)T+(6.85e3+3.96e3i)T2 1 + (-85.6 - 22.9i)T + (6.85e3 + 3.96e3i)T^{2}
97 1+(32.6−8.75i)T+(8.14e3−4.70e3i)T2 1 + (32.6 - 8.75i)T + (8.14e3 - 4.70e3i)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

−13.07103363967669989999479787732, −11.80724639404624003359222404248, −10.72590364964405311703242335589, −9.631693013635334598130668962226, −8.921276875808931986782856611631, −7.27393846156734682353893206190, −6.60661573277273417416939891517, −5.95526555596792784850381365019, −3.92691926232441695487836156234, −2.26053475659519885743776778064, 1.32284540291596326444219357799, 2.25872887536294057010202119139, 4.24998721295863761627849222269, 5.72106023101814798084479483230, 6.76924379550929555669288640842, 8.709553088459899940092482499455, 9.174014490529345492707623716000, 10.26370459682319378281878868735, 11.12409032943640179315681185309, 12.35498314887585470035288333655

Graph of the ZZ-function along the critical line