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Properties

Label 2-65-13.12-c5-0-7
Degree 22
Conductor 6565
Sign 0.631−0.775i0.631 - 0.775i
Analytic cond. 10.424910.4249
Root an. cond. 3.228763.22876
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.21i·2-s − 11.3·3-s + 30.5·4-s + 25i·5-s + 13.8i·6-s − 6.14i·7-s − 76.2i·8-s − 113.·9-s + 30.4·10-s + 391. i·11-s − 347.·12-s + (472. + 384. i)13-s − 7.48·14-s − 284. i·15-s + 883.·16-s + 1.54e3·17-s + ⋯
L(s)  = 1  − 0.215i·2-s − 0.729·3-s + 0.953·4-s + 0.447i·5-s + 0.157i·6-s − 0.0473i·7-s − 0.421i·8-s − 0.467·9-s + 0.0963·10-s + 0.974i·11-s − 0.695·12-s + (0.775 + 0.631i)13-s − 0.0102·14-s − 0.326i·15-s + 0.862·16-s + 1.29·17-s + ⋯

Functional equation

Λ(s)=(65s/2 ΓC(s) L(s)=((0.631−0.775i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(65s/2 ΓC(s+5/2) L(s)=((0.631−0.775i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6565    =    5⋅135 \cdot 13
Sign: 0.631−0.775i0.631 - 0.775i
Analytic conductor: 10.424910.4249
Root analytic conductor: 3.228763.22876
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ65(51,⋅)\chi_{65} (51, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 65, ( :5/2), 0.631−0.775i)(2,\ 65,\ (\ :5/2),\ 0.631 - 0.775i)

Particular Values

L(3)L(3) ≈\approx 1.40632+0.668348i1.40632 + 0.668348i
L(12)L(\frac12) ≈\approx 1.40632+0.668348i1.40632 + 0.668348i
L(72)L(\frac{7}{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)
bad5 1−25iT 1 - 25iT
13 1+(−472.−384.i)T 1 + (-472. - 384. i)T
good2 1+1.21iT−32T2 1 + 1.21iT - 32T^{2}
3 1+11.3T+243T2 1 + 11.3T + 243T^{2}
7 1+6.14iT−1.68e4T2 1 + 6.14iT - 1.68e4T^{2}
11 1−391.iT−1.61e5T2 1 - 391. iT - 1.61e5T^{2}
17 1−1.54e3T+1.41e6T2 1 - 1.54e3T + 1.41e6T^{2}
19 1−3.01e3iT−2.47e6T2 1 - 3.01e3iT - 2.47e6T^{2}
23 1−921.T+6.43e6T2 1 - 921.T + 6.43e6T^{2}
29 1+5.24e3T+2.05e7T2 1 + 5.24e3T + 2.05e7T^{2}
31 1−1.74e3iT−2.86e7T2 1 - 1.74e3iT - 2.86e7T^{2}
37 1−2.78e3iT−6.93e7T2 1 - 2.78e3iT - 6.93e7T^{2}
41 1−7.30e3iT−1.15e8T2 1 - 7.30e3iT - 1.15e8T^{2}
43 1+2.46e3T+1.47e8T2 1 + 2.46e3T + 1.47e8T^{2}
47 1+2.26e4iT−2.29e8T2 1 + 2.26e4iT - 2.29e8T^{2}
53 1+2.73e4T+4.18e8T2 1 + 2.73e4T + 4.18e8T^{2}
59 1+4.76e4iT−7.14e8T2 1 + 4.76e4iT - 7.14e8T^{2}
61 1+2.58e4T+8.44e8T2 1 + 2.58e4T + 8.44e8T^{2}
67 1−3.62e4iT−1.35e9T2 1 - 3.62e4iT - 1.35e9T^{2}
71 1−2.52e4iT−1.80e9T2 1 - 2.52e4iT - 1.80e9T^{2}
73 1+6.05e4iT−2.07e9T2 1 + 6.05e4iT - 2.07e9T^{2}
79 1−8.76e4T+3.07e9T2 1 - 8.76e4T + 3.07e9T^{2}
83 1−8.24e4iT−3.93e9T2 1 - 8.24e4iT - 3.93e9T^{2}
89 1−2.60e4iT−5.58e9T2 1 - 2.60e4iT - 5.58e9T^{2}
97 1+1.35e5iT−8.58e9T2 1 + 1.35e5iT - 8.58e9T^{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

−14.23384085225799150989236604811, −12.48367334442932776369317706085, −11.76682285133220820503950232774, −10.81066140275556304489173877833, −9.860256434831619662488498393830, −7.894631987468914639427366702717, −6.64043896633120461430525062415, −5.61905341884784790673032839397, −3.49221941169252596560477264163, −1.64100649405658500953760186325, 0.820953959190761149791128661615, 3.05491424202200746819560979093, 5.37670275417430131767769711339, 6.15492354417010067842556343743, 7.65895551500918884874234207928, 8.946549399489030485128860341445, 10.79120380616896912831256556840, 11.31900200490332457983006735704, 12.42708618866790674053924275693, 13.70434539706976184419368830902

Graph of the ZZ-function along the critical line