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Properties

Label 2-26-13.3-c3-0-0
Degree $2$
Conductor $26$
Sign $0.937 - 0.348i$
Analytic cond. $1.53404$
Root an. cond. $1.23856$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (4.43 + 7.67i)3-s + (−1.99 + 3.46i)4-s + 3.86·5-s + (8.86 − 15.3i)6-s + (7.56 − 13.1i)7-s + 7.99·8-s + (−25.7 + 44.6i)9-s + (−3.86 − 6.69i)10-s + (−27.0 − 46.8i)11-s − 35.4·12-s + (−19.7 + 42.4i)13-s − 30.2·14-s + (17.1 + 29.6i)15-s + (−8 − 13.8i)16-s + (61.9 − 107. i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.853 + 1.47i)3-s + (−0.249 + 0.433i)4-s + 0.345·5-s + (0.603 − 1.04i)6-s + (0.408 − 0.707i)7-s + 0.353·8-s + (−0.955 + 1.65i)9-s + (−0.122 − 0.211i)10-s + (−0.740 − 1.28i)11-s − 0.853·12-s + (−0.422 + 0.906i)13-s − 0.577·14-s + (0.294 + 0.510i)15-s + (−0.125 − 0.216i)16-s + (0.883 − 1.53i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.348i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $0.937 - 0.348i$
Analytic conductor: \(1.53404\)
Root analytic conductor: \(1.23856\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{26} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 26,\ (\ :3/2),\ 0.937 - 0.348i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.21934 + 0.219197i\)
\(L(\frac12)\) \(\approx\) \(1.21934 + 0.219197i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
13 \( 1 + (19.7 - 42.4i)T \)
good3 \( 1 + (-4.43 - 7.67i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 3.86T + 125T^{2} \)
7 \( 1 + (-7.56 + 13.1i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (27.0 + 46.8i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-61.9 + 107. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (4.43 - 7.67i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (19.2 + 33.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-93.6 - 162. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 36.7T + 2.97e4T^{2} \)
37 \( 1 + (-160. - 278. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-13.1 - 22.7i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (68.8 - 119. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 300.T + 1.03e5T^{2} \)
53 \( 1 + 260.T + 1.48e5T^{2} \)
59 \( 1 + (-123. + 213. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (45.6 - 79.0i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (205. + 355. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-212. + 367. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 421.T + 3.89e5T^{2} \)
79 \( 1 - 733.T + 4.93e5T^{2} \)
83 \( 1 + 616.T + 5.71e5T^{2} \)
89 \( 1 + (-103. - 179. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-370. + 642. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−16.73112378854812163131013349023, −16.07057883600783454144399520295, −14.33624431441409987445945376478, −13.70367260191904259767813895005, −11.41298565846761534169495719962, −10.28323222592373595737720413344, −9.347014839264280800319294706777, −8.015176488340728802219689962142, −4.80128448677247860820964192422, −3.11491325376961030866149668868, 2.06980843022981871750615492600, 5.82533375931178049587334571822, 7.52447581380564078375526246085, 8.274971565934567641208667797460, 9.921244768777208482516415991028, 12.30633046870530535484613056097, 13.15564382592871775920621335707, 14.56459927741004992978303773182, 15.29489146617131036351877985056, 17.50458258406368244486365528851

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