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Properties

Label 2-26-13.3-c3-0-0
Degree 22
Conductor 2626
Sign 0.937−0.348i0.937 - 0.348i
Analytic cond. 1.534041.53404
Root an. cond. 1.238561.23856
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (4.43 + 7.67i)3-s + (−1.99 + 3.46i)4-s + 3.86·5-s + (8.86 − 15.3i)6-s + (7.56 − 13.1i)7-s + 7.99·8-s + (−25.7 + 44.6i)9-s + (−3.86 − 6.69i)10-s + (−27.0 − 46.8i)11-s − 35.4·12-s + (−19.7 + 42.4i)13-s − 30.2·14-s + (17.1 + 29.6i)15-s + (−8 − 13.8i)16-s + (61.9 − 107. i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.853 + 1.47i)3-s + (−0.249 + 0.433i)4-s + 0.345·5-s + (0.603 − 1.04i)6-s + (0.408 − 0.707i)7-s + 0.353·8-s + (−0.955 + 1.65i)9-s + (−0.122 − 0.211i)10-s + (−0.740 − 1.28i)11-s − 0.853·12-s + (−0.422 + 0.906i)13-s − 0.577·14-s + (0.294 + 0.510i)15-s + (−0.125 − 0.216i)16-s + (0.883 − 1.53i)17-s + ⋯

Functional equation

Λ(s)=(26s/2 ΓC(s) L(s)=((0.937−0.348i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.348i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(26s/2 ΓC(s+3/2) L(s)=((0.937−0.348i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2626    =    2⋅132 \cdot 13
Sign: 0.937−0.348i0.937 - 0.348i
Analytic conductor: 1.534041.53404
Root analytic conductor: 1.238561.23856
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ26(3,⋅)\chi_{26} (3, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 26, ( :3/2), 0.937−0.348i)(2,\ 26,\ (\ :3/2),\ 0.937 - 0.348i)

Particular Values

L(2)L(2) ≈\approx 1.21934+0.219197i1.21934 + 0.219197i
L(12)L(\frac12) ≈\approx 1.21934+0.219197i1.21934 + 0.219197i
L(52)L(\frac{5}{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)
bad2 1+(1+1.73i)T 1 + (1 + 1.73i)T
13 1+(19.7−42.4i)T 1 + (19.7 - 42.4i)T
good3 1+(−4.43−7.67i)T+(−13.5+23.3i)T2 1 + (-4.43 - 7.67i)T + (-13.5 + 23.3i)T^{2}
5 1−3.86T+125T2 1 - 3.86T + 125T^{2}
7 1+(−7.56+13.1i)T+(−171.5−297.i)T2 1 + (-7.56 + 13.1i)T + (-171.5 - 297. i)T^{2}
11 1+(27.0+46.8i)T+(−665.5+1.15e3i)T2 1 + (27.0 + 46.8i)T + (-665.5 + 1.15e3i)T^{2}
17 1+(−61.9+107.i)T+(−2.45e3−4.25e3i)T2 1 + (-61.9 + 107. i)T + (-2.45e3 - 4.25e3i)T^{2}
19 1+(4.43−7.67i)T+(−3.42e3−5.94e3i)T2 1 + (4.43 - 7.67i)T + (-3.42e3 - 5.94e3i)T^{2}
23 1+(19.2+33.4i)T+(−6.08e3+1.05e4i)T2 1 + (19.2 + 33.4i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1+(−93.6−162.i)T+(−1.21e4+2.11e4i)T2 1 + (-93.6 - 162. i)T + (-1.21e4 + 2.11e4i)T^{2}
31 1+36.7T+2.97e4T2 1 + 36.7T + 2.97e4T^{2}
37 1+(−160.−278.i)T+(−2.53e4+4.38e4i)T2 1 + (-160. - 278. i)T + (-2.53e4 + 4.38e4i)T^{2}
41 1+(−13.1−22.7i)T+(−3.44e4+5.96e4i)T2 1 + (-13.1 - 22.7i)T + (-3.44e4 + 5.96e4i)T^{2}
43 1+(68.8−119.i)T+(−3.97e4−6.88e4i)T2 1 + (68.8 - 119. i)T + (-3.97e4 - 6.88e4i)T^{2}
47 1+300.T+1.03e5T2 1 + 300.T + 1.03e5T^{2}
53 1+260.T+1.48e5T2 1 + 260.T + 1.48e5T^{2}
59 1+(−123.+213.i)T+(−1.02e5−1.77e5i)T2 1 + (-123. + 213. i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(45.6−79.0i)T+(−1.13e5−1.96e5i)T2 1 + (45.6 - 79.0i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(205.+355.i)T+(−1.50e5+2.60e5i)T2 1 + (205. + 355. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1+(−212.+367.i)T+(−1.78e5−3.09e5i)T2 1 + (-212. + 367. i)T + (-1.78e5 - 3.09e5i)T^{2}
73 1+421.T+3.89e5T2 1 + 421.T + 3.89e5T^{2}
79 1−733.T+4.93e5T2 1 - 733.T + 4.93e5T^{2}
83 1+616.T+5.71e5T2 1 + 616.T + 5.71e5T^{2}
89 1+(−103.−179.i)T+(−3.52e5+6.10e5i)T2 1 + (-103. - 179. i)T + (-3.52e5 + 6.10e5i)T^{2}
97 1+(−370.+642.i)T+(−4.56e5−7.90e5i)T2 1 + (-370. + 642. i)T + (-4.56e5 - 7.90e5i)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

−16.73112378854812163131013349023, −16.07057883600783454144399520295, −14.33624431441409987445945376478, −13.70367260191904259767813895005, −11.41298565846761534169495719962, −10.28323222592373595737720413344, −9.347014839264280800319294706777, −8.015176488340728802219689962142, −4.80128448677247860820964192422, −3.11491325376961030866149668868, 2.06980843022981871750615492600, 5.82533375931178049587334571822, 7.52447581380564078375526246085, 8.274971565934567641208667797460, 9.921244768777208482516415991028, 12.30633046870530535484613056097, 13.15564382592871775920621335707, 14.56459927741004992978303773182, 15.29489146617131036351877985056, 17.50458258406368244486365528851

Graph of the ZZ-function along the critical line