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Properties

Label 2-26-13.2-c4-0-2
Degree 22
Conductor 2626
Sign 0.713−0.700i0.713 - 0.700i
Analytic cond. 2.687612.68761
Root an. cond. 1.639391.63939
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.73 + 0.732i)2-s + (3.73 + 6.46i)3-s + (6.92 + 4i)4-s + (−3.43 + 3.43i)5-s + (5.46 + 20.3i)6-s + (5.09 − 1.36i)7-s + (15.9 + 16i)8-s + (12.6 − 21.9i)9-s + (−11.9 + 6.87i)10-s + (18.2 − 68.0i)11-s + 59.7i·12-s + (−147. − 82.1i)13-s + 14.9·14-s + (−35.0 − 9.38i)15-s + (31.9 + 55.4i)16-s + (−163. − 94.5i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.414 + 0.718i)3-s + (0.433 + 0.250i)4-s + (−0.137 + 0.137i)5-s + (0.151 + 0.566i)6-s + (0.104 − 0.0278i)7-s + (0.249 + 0.250i)8-s + (0.156 − 0.270i)9-s + (−0.119 + 0.0687i)10-s + (0.150 − 0.562i)11-s + 0.414i·12-s + (−0.873 − 0.486i)13-s + 0.0761·14-s + (−0.155 − 0.0417i)15-s + (0.124 + 0.216i)16-s + (−0.566 − 0.327i)17-s + ⋯

Functional equation

Λ(s)=(26s/2 ΓC(s) L(s)=((0.713−0.700i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(26s/2 ΓC(s+2) L(s)=((0.713−0.700i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2626    =    2⋅132 \cdot 13
Sign: 0.713−0.700i0.713 - 0.700i
Analytic conductor: 2.687612.68761
Root analytic conductor: 1.639391.63939
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ26(15,⋅)\chi_{26} (15, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 26, ( :2), 0.713−0.700i)(2,\ 26,\ (\ :2),\ 0.713 - 0.700i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 1.90328+0.777510i1.90328 + 0.777510i
L(12)L(\frac12) ≈\approx 1.90328+0.777510i1.90328 + 0.777510i
L(3)L(3) 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+(−2.73−0.732i)T 1 + (-2.73 - 0.732i)T
13 1+(147.+82.1i)T 1 + (147. + 82.1i)T
good3 1+(−3.73−6.46i)T+(−40.5+70.1i)T2 1 + (-3.73 - 6.46i)T + (-40.5 + 70.1i)T^{2}
5 1+(3.43−3.43i)T−625iT2 1 + (3.43 - 3.43i)T - 625iT^{2}
7 1+(−5.09+1.36i)T+(2.07e3−1.20e3i)T2 1 + (-5.09 + 1.36i)T + (2.07e3 - 1.20e3i)T^{2}
11 1+(−18.2+68.0i)T+(−1.26e4−7.32e3i)T2 1 + (-18.2 + 68.0i)T + (-1.26e4 - 7.32e3i)T^{2}
17 1+(163.+94.5i)T+(4.17e4+7.23e4i)T2 1 + (163. + 94.5i)T + (4.17e4 + 7.23e4i)T^{2}
19 1+(51.3+191.i)T+(−1.12e5+6.51e4i)T2 1 + (51.3 + 191. i)T + (-1.12e5 + 6.51e4i)T^{2}
23 1+(27.6−15.9i)T+(1.39e5−2.42e5i)T2 1 + (27.6 - 15.9i)T + (1.39e5 - 2.42e5i)T^{2}
29 1+(−724.−1.25e3i)T+(−3.53e5+6.12e5i)T2 1 + (-724. - 1.25e3i)T + (-3.53e5 + 6.12e5i)T^{2}
31 1+(23.2−23.2i)T−9.23e5iT2 1 + (23.2 - 23.2i)T - 9.23e5iT^{2}
37 1+(203.−758.i)T+(−1.62e6−9.37e5i)T2 1 + (203. - 758. i)T + (-1.62e6 - 9.37e5i)T^{2}
41 1+(439.+117.i)T+(2.44e6+1.41e6i)T2 1 + (439. + 117. i)T + (2.44e6 + 1.41e6i)T^{2}
43 1+(2.51e3+1.45e3i)T+(1.70e6+2.96e6i)T2 1 + (2.51e3 + 1.45e3i)T + (1.70e6 + 2.96e6i)T^{2}
47 1+(1.37e3+1.37e3i)T+4.87e6iT2 1 + (1.37e3 + 1.37e3i)T + 4.87e6iT^{2}
53 1−3.45e3T+7.89e6T2 1 - 3.45e3T + 7.89e6T^{2}
59 1+(−6.20e3+1.66e3i)T+(1.04e7−6.05e6i)T2 1 + (-6.20e3 + 1.66e3i)T + (1.04e7 - 6.05e6i)T^{2}
61 1+(3.10e3−5.38e3i)T+(−6.92e6−1.19e7i)T2 1 + (3.10e3 - 5.38e3i)T + (-6.92e6 - 1.19e7i)T^{2}
67 1+(2.78e3+745.i)T+(1.74e7+1.00e7i)T2 1 + (2.78e3 + 745. i)T + (1.74e7 + 1.00e7i)T^{2}
71 1+(−791.−2.95e3i)T+(−2.20e7+1.27e7i)T2 1 + (-791. - 2.95e3i)T + (-2.20e7 + 1.27e7i)T^{2}
73 1+(4.76e3+4.76e3i)T+2.83e7iT2 1 + (4.76e3 + 4.76e3i)T + 2.83e7iT^{2}
79 1−2.82e3T+3.89e7T2 1 - 2.82e3T + 3.89e7T^{2}
83 1+(−7.46e3+7.46e3i)T−4.74e7iT2 1 + (-7.46e3 + 7.46e3i)T - 4.74e7iT^{2}
89 1+(−979.+3.65e3i)T+(−5.43e7−3.13e7i)T2 1 + (-979. + 3.65e3i)T + (-5.43e7 - 3.13e7i)T^{2}
97 1+(−63.2−235.i)T+(−7.66e7+4.42e7i)T2 1 + (-63.2 - 235. i)T + (-7.66e7 + 4.42e7i)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

−16.48045627423468707044700866795, −15.35086211030961794288275611256, −14.60380878242848531038501878414, −13.25496612854472803718676387218, −11.80821258113160873964517010102, −10.33986849750646214541612179771, −8.783873625917649147030368291222, −6.94552622341130853083284833195, −4.92955580293316892419119991218, −3.26056089029419311340724104468, 2.14174594879181511453489401984, 4.55143479834713795941466800194, 6.68139429549154780970785242807, 8.088145575041939862798703442861, 10.05751729070934813465558439209, 11.80096834605982323192265366161, 12.80698607498361160185442452159, 13.92043326473997408259952418507, 14.98501440330062831964032861295, 16.40149341605685704625152749174

Graph of the ZZ-function along the critical line