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Properties

Label 2-26-13.6-c2-0-2
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: $\chi_{26} (19, \cdot )$
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

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

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