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Properties

Label 2-26-13.11-c2-0-0
Degree $2$
Conductor $26$
Sign $0.538 - 0.842i$
Analytic cond. $0.708448$
Root an. cond. $0.841693$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $0.538 - 0.842i$
Analytic conductor: \(0.708448\)
Root analytic conductor: \(0.841693\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{26} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 26,\ (\ :1),\ 0.538 - 0.842i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.801043 + 0.438742i\)
\(L(\frac12)\) \(\approx\) \(0.801043 + 0.438742i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

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

−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 $Z$-function along the critical line