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Properties

Label 2-26-13.8-c4-0-5
Degree $2$
Conductor $26$
Sign $-0.999 + 0.0195i$
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 − 2.19·3-s + 8i·4-s + (−28.1 − 28.1i)5-s + (4.38 + 4.38i)6-s + (−49.0 + 49.0i)7-s + (16 − 16i)8-s − 76.1·9-s + 112. i·10-s + (125. − 125. i)11-s − 17.5i·12-s + (76.8 − 150. i)13-s + 196.·14-s + (61.6 + 61.6i)15-s − 64·16-s + 293. i·17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s − 0.243·3-s + 0.5i·4-s + (−1.12 − 1.12i)5-s + (0.121 + 0.121i)6-s + (−1.00 + 1.00i)7-s + (0.250 − 0.250i)8-s − 0.940·9-s + 1.12i·10-s + (1.04 − 1.04i)11-s − 0.121i·12-s + (0.454 − 0.890i)13-s + 1.00·14-s + (0.273 + 0.273i)15-s − 0.250·16-s + 1.01i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0195i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $-0.999 + 0.0195i$
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: \((2,\ 26,\ (\ :2),\ -0.999 + 0.0195i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.00292541 - 0.298842i\)
\(L(\frac12)\) \(\approx\) \(0.00292541 - 0.298842i\)
\(L(3)\) 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 \( 1 + (-76.8 + 150. i)T \)
good3 \( 1 + 2.19T + 81T^{2} \)
5 \( 1 + (28.1 + 28.1i)T + 625iT^{2} \)
7 \( 1 + (49.0 - 49.0i)T - 2.40e3iT^{2} \)
11 \( 1 + (-125. + 125. i)T - 1.46e4iT^{2} \)
17 \( 1 - 293. iT - 8.35e4T^{2} \)
19 \( 1 + (165. + 165. i)T + 1.30e5iT^{2} \)
23 \( 1 + 297. iT - 2.79e5T^{2} \)
29 \( 1 + 1.06e3T + 7.07e5T^{2} \)
31 \( 1 + (511. + 511. i)T + 9.23e5iT^{2} \)
37 \( 1 + (292. - 292. i)T - 1.87e6iT^{2} \)
41 \( 1 + (804. + 804. i)T + 2.82e6iT^{2} \)
43 \( 1 - 1.00e3iT - 3.41e6T^{2} \)
47 \( 1 + (-1.32e3 + 1.32e3i)T - 4.87e6iT^{2} \)
53 \( 1 - 574.T + 7.89e6T^{2} \)
59 \( 1 + (-2.23e3 + 2.23e3i)T - 1.21e7iT^{2} \)
61 \( 1 + 6.06e3T + 1.38e7T^{2} \)
67 \( 1 + (-3.15e3 - 3.15e3i)T + 2.01e7iT^{2} \)
71 \( 1 + (1.60e3 + 1.60e3i)T + 2.54e7iT^{2} \)
73 \( 1 + (-22.6 + 22.6i)T - 2.83e7iT^{2} \)
79 \( 1 - 6.57e3T + 3.89e7T^{2} \)
83 \( 1 + (5.69e3 + 5.69e3i)T + 4.74e7iT^{2} \)
89 \( 1 + (-4.24e3 + 4.24e3i)T - 6.27e7iT^{2} \)
97 \( 1 + (-3.53e3 - 3.53e3i)T + 8.85e7iT^{2} \)
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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

−16.36696219385507819899357527323, −15.19160454004087618209667613224, −13.00358412212652265201208997495, −12.10976421793087286265184805786, −11.10556132697993395492217767736, −8.991020950298348961633515543839, −8.406161020333095979480637157662, −5.91716553183966556323648521554, −3.54227916126587490346538083060, −0.28231814727009740183801353445, 3.78373725725194080129106573660, 6.59478561375411852957053132514, 7.36386223469181616892885471403, 9.338553122131055608030517916315, 10.81122715752117207316430944503, 11.86619517995802716239209411872, 13.97050127358861502058716729300, 14.94174038756342905714241159081, 16.23741461949377382831972651034, 17.10945835476318134654698205439

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