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Properties

Label 2-26-13.8-c4-0-3
Degree $2$
Conductor $26$
Sign $0.824 + 0.565i$
Analytic cond. $2.68761$
Root an. cond. $1.63939$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 2i)2-s + 11.7·3-s + 8i·4-s + (3.61 + 3.61i)5-s + (−23.4 − 23.4i)6-s + (52.3 − 52.3i)7-s + (16 − 16i)8-s + 56.9·9-s − 14.4i·10-s + (−28.9 + 28.9i)11-s + 93.9i·12-s + (−150. + 77.7i)13-s − 209.·14-s + (42.3 + 42.3i)15-s − 64·16-s + 451. i·17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + 1.30·3-s + 0.5i·4-s + (0.144 + 0.144i)5-s + (−0.652 − 0.652i)6-s + (1.06 − 1.06i)7-s + (0.250 − 0.250i)8-s + 0.703·9-s − 0.144i·10-s + (−0.239 + 0.239i)11-s + 0.652i·12-s + (−0.887 + 0.460i)13-s − 1.06·14-s + (0.188 + 0.188i)15-s − 0.250·16-s + 1.56i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $0.824 + 0.565i$
Analytic conductor: \(2.68761\)
Root analytic conductor: \(1.63939\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{26} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 26,\ (\ :2),\ 0.824 + 0.565i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.49477 - 0.463344i\)
\(L(\frac12)\) \(\approx\) \(1.49477 - 0.463344i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 + 2i)T \)
13 \( 1 + (150. - 77.7i)T \)
good3 \( 1 - 11.7T + 81T^{2} \)
5 \( 1 + (-3.61 - 3.61i)T + 625iT^{2} \)
7 \( 1 + (-52.3 + 52.3i)T - 2.40e3iT^{2} \)
11 \( 1 + (28.9 - 28.9i)T - 1.46e4iT^{2} \)
17 \( 1 - 451. iT - 8.35e4T^{2} \)
19 \( 1 + (164. + 164. i)T + 1.30e5iT^{2} \)
23 \( 1 - 268. iT - 2.79e5T^{2} \)
29 \( 1 + 1.24e3T + 7.07e5T^{2} \)
31 \( 1 + (-511. - 511. i)T + 9.23e5iT^{2} \)
37 \( 1 + (-1.70e3 + 1.70e3i)T - 1.87e6iT^{2} \)
41 \( 1 + (923. + 923. i)T + 2.82e6iT^{2} \)
43 \( 1 - 1.85e3iT - 3.41e6T^{2} \)
47 \( 1 + (689. - 689. i)T - 4.87e6iT^{2} \)
53 \( 1 - 2.82e3T + 7.89e6T^{2} \)
59 \( 1 + (-1.53e3 + 1.53e3i)T - 1.21e7iT^{2} \)
61 \( 1 - 4.89e3T + 1.38e7T^{2} \)
67 \( 1 + (4.46e3 + 4.46e3i)T + 2.01e7iT^{2} \)
71 \( 1 + (-6.33e3 - 6.33e3i)T + 2.54e7iT^{2} \)
73 \( 1 + (2.30e3 - 2.30e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 6.46e3T + 3.89e7T^{2} \)
83 \( 1 + (5.15e3 + 5.15e3i)T + 4.74e7iT^{2} \)
89 \( 1 + (1.75e3 - 1.75e3i)T - 6.27e7iT^{2} \)
97 \( 1 + (4.09e3 + 4.09e3i)T + 8.85e7iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.90762201823824432067853800737, −14.95995865730580231318038069371, −14.17541784300545121888409716050, −12.93096219212733205552747065309, −11.11476007994746878025609162446, −9.879586750597190775709061783160, −8.433031923027220567466034043014, −7.43999989611025829924786281089, −4.10425854627304862289255746743, −2.04846569337037926036947322826, 2.39866890956647433611228462689, 5.26307206046254425364284506853, 7.63419448498928049260858427720, 8.601166585153274521895307582034, 9.647723723692970085755233434453, 11.61123203253415548987367889359, 13.42874805464805592398032098882, 14.74684300342377816496455710054, 15.16138946044613151209171667906, 16.80664778238365284527461819492

Graph of the $Z$-function along the critical line