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Properties

Label 2-26-13.11-c2-0-0
Degree 22
Conductor 2626
Sign 0.538−0.842i0.538 - 0.842i
Analytic cond. 0.7084480.708448
Root an. cond. 0.8416930.841693
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  + (−0.366 + 1.36i)2-s + (0.866 + 1.5i)3-s + (−1.73 − i)4-s + (1.73 + 1.73i)5-s + (−2.36 + 0.633i)6-s + (−2.03 − 7.59i)7-s + (2 − 1.99i)8-s + (3 − 5.19i)9-s + (−2.99 + 1.73i)10-s + (−4.96 − 1.33i)11-s − 3.46i·12-s + (−9.92 + 8.39i)13-s + 11.1·14-s + (−1.09 + 4.09i)15-s + (1.99 + 3.46i)16-s + (24.6 + 14.2i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.288 + 0.5i)3-s + (−0.433 − 0.250i)4-s + (0.346 + 0.346i)5-s + (−0.394 + 0.105i)6-s + (−0.290 − 1.08i)7-s + (0.250 − 0.249i)8-s + (0.333 − 0.577i)9-s + (−0.299 + 0.173i)10-s + (−0.451 − 0.120i)11-s − 0.288i·12-s + (−0.763 + 0.645i)13-s + 0.794·14-s + (−0.0732 + 0.273i)15-s + (0.124 + 0.216i)16-s + (1.45 + 0.838i)17-s + ⋯

Functional equation

Λ(s)=(26s/2 ΓC(s) L(s)=((0.538−0.842i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(26s/2 ΓC(s+1) L(s)=((0.538−0.842i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2626    =    2⋅132 \cdot 13
Sign: 0.538−0.842i0.538 - 0.842i
Analytic conductor: 0.7084480.708448
Root analytic conductor: 0.8416930.841693
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ26(11,⋅)\chi_{26} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 26, ( :1), 0.538−0.842i)(2,\ 26,\ (\ :1),\ 0.538 - 0.842i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.801043+0.438742i0.801043 + 0.438742i
L(12)L(\frac12) ≈\approx 0.801043+0.438742i0.801043 + 0.438742i
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)
bad2 1+(0.366−1.36i)T 1 + (0.366 - 1.36i)T
13 1+(9.92−8.39i)T 1 + (9.92 - 8.39i)T
good3 1+(−0.866−1.5i)T+(−4.5+7.79i)T2 1 + (-0.866 - 1.5i)T + (-4.5 + 7.79i)T^{2}
5 1+(−1.73−1.73i)T+25iT2 1 + (-1.73 - 1.73i)T + 25iT^{2}
7 1+(2.03+7.59i)T+(−42.4+24.5i)T2 1 + (2.03 + 7.59i)T + (-42.4 + 24.5i)T^{2}
11 1+(4.96+1.33i)T+(104.+60.5i)T2 1 + (4.96 + 1.33i)T + (104. + 60.5i)T^{2}
17 1+(−24.6−14.2i)T+(144.5+250.i)T2 1 + (-24.6 - 14.2i)T + (144.5 + 250. i)T^{2}
19 1+(24.8−6.66i)T+(312.−180.5i)T2 1 + (24.8 - 6.66i)T + (312. - 180.5i)T^{2}
23 1+(15.1−8.76i)T+(264.5−458.i)T2 1 + (15.1 - 8.76i)T + (264.5 - 458. i)T^{2}
29 1+(−0.356−0.617i)T+(−420.5+728.i)T2 1 + (-0.356 - 0.617i)T + (-420.5 + 728. i)T^{2}
31 1+(23.3+23.3i)T+961iT2 1 + (23.3 + 23.3i)T + 961iT^{2}
37 1+(−21.2−5.69i)T+(1.18e3+684.5i)T2 1 + (-21.2 - 5.69i)T + (1.18e3 + 684.5i)T^{2}
41 1+(11.2−42.0i)T+(−1.45e3−840.5i)T2 1 + (11.2 - 42.0i)T + (-1.45e3 - 840.5i)T^{2}
43 1+(−63.7−36.8i)T+(924.5+1.60e3i)T2 1 + (-63.7 - 36.8i)T + (924.5 + 1.60e3i)T^{2}
47 1+(−33+33i)T−2.20e3iT2 1 + (-33 + 33i)T - 2.20e3iT^{2}
53 1+80.1T+2.80e3T2 1 + 80.1T + 2.80e3T^{2}
59 1+(−10.1−37.9i)T+(−3.01e3+1.74e3i)T2 1 + (-10.1 - 37.9i)T + (-3.01e3 + 1.74e3i)T^{2}
61 1+(−14.3+24.7i)T+(−1.86e3−3.22e3i)T2 1 + (-14.3 + 24.7i)T + (-1.86e3 - 3.22e3i)T^{2}
67 1+(−12.8+47.9i)T+(−3.88e3−2.24e3i)T2 1 + (-12.8 + 47.9i)T + (-3.88e3 - 2.24e3i)T^{2}
71 1+(−7.03+1.88i)T+(4.36e3−2.52e3i)T2 1 + (-7.03 + 1.88i)T + (4.36e3 - 2.52e3i)T^{2}
73 1+(−12.7+12.7i)T−5.32e3iT2 1 + (-12.7 + 12.7i)T - 5.32e3iT^{2}
79 1+14.3T+6.24e3T2 1 + 14.3T + 6.24e3T^{2}
83 1+(−87.8−87.8i)T+6.88e3iT2 1 + (-87.8 - 87.8i)T + 6.88e3iT^{2}
89 1+(52.2+13.9i)T+(6.85e3+3.96e3i)T2 1 + (52.2 + 13.9i)T + (6.85e3 + 3.96e3i)T^{2}
97 1+(−131.+35.2i)T+(8.14e3−4.70e3i)T2 1 + (-131. + 35.2i)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

−17.16681316004595320679292317531, −16.35340360051428110843122838895, −14.91153427385912207106049339107, −14.15645822703157626923880970090, −12.66271162178600976518060930701, −10.43282061106278180731677964546, −9.641565124467089165049291160972, −7.80562933100406125597898926348, −6.31424320090316127341430914194, −4.07335077191326694410777982992, 2.44872389725305077785373347331, 5.31972203220906167945160620566, 7.69368741356450004548514797437, 9.129428276412286806304527746000, 10.45145477118621779010823865466, 12.30987150271449776107319815442, 12.89145518612938970727692159288, 14.35103853744856204047209784366, 15.91411593938446182523645995496, 17.33856685058084169175103324812

Graph of the ZZ-function along the critical line