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Properties

Label 2-26-13.10-c3-0-3
Degree $2$
Conductor $26$
Sign $0.244 + 0.969i$
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  + (1.73 − i)2-s + (−3.93 − 6.82i)3-s + (1.99 − 3.46i)4-s + 3.30i·5-s + (−13.6 − 7.87i)6-s + (27.1 + 15.6i)7-s − 7.99i·8-s + (−17.5 + 30.3i)9-s + (3.30 + 5.72i)10-s + (13.5 − 7.80i)11-s − 31.5·12-s + (−45.2 − 12.2i)13-s + 62.7·14-s + (22.5 − 13.0i)15-s + (−8 − 13.8i)16-s + (−53.8 + 93.2i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.757 − 1.31i)3-s + (0.249 − 0.433i)4-s + 0.295i·5-s + (−0.928 − 0.535i)6-s + (1.46 + 0.847i)7-s − 0.353i·8-s + (−0.648 + 1.12i)9-s + (0.104 + 0.181i)10-s + (0.370 − 0.213i)11-s − 0.757·12-s + (−0.965 − 0.260i)13-s + 1.19·14-s + (0.388 − 0.224i)15-s + (−0.125 − 0.216i)16-s + (−0.768 + 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.244 + 0.969i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $0.244 + 0.969i$
Analytic conductor: \(1.53404\)
Root analytic conductor: \(1.23856\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{26} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 26,\ (\ :3/2),\ 0.244 + 0.969i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.07551 - 0.838075i\)
\(L(\frac12)\) \(\approx\) \(1.07551 - 0.838075i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.73 + i)T \)
13 \( 1 + (45.2 + 12.2i)T \)
good3 \( 1 + (3.93 + 6.82i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 3.30iT - 125T^{2} \)
7 \( 1 + (-27.1 - 15.6i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-13.5 + 7.80i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (53.8 - 93.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-52.6 - 30.3i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (62.1 + 107. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (29.1 + 50.4i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 200. iT - 2.97e4T^{2} \)
37 \( 1 + (90.9 - 52.5i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (191. - 110. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-56.0 + 97.0i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 512. iT - 1.03e5T^{2} \)
53 \( 1 + 221.T + 1.48e5T^{2} \)
59 \( 1 + (482. + 278. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-229. + 396. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-458. + 264. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-58.5 - 33.7i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 104. iT - 3.89e5T^{2} \)
79 \( 1 - 611.T + 4.93e5T^{2} \)
83 \( 1 - 491. iT - 5.71e5T^{2} \)
89 \( 1 + (-326. + 188. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (496. + 286. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−17.11253226967435358479314307688, −15.07526386834331592777356934036, −14.11620808780201996721979729202, −12.56631071186745854762183705014, −11.89339979386038912590231776909, −10.79817218959722222193822344186, −8.224986110142498035863455076317, −6.60010098708107718570133843390, −5.17822042389354457891908704054, −1.89095595527684902840214352272, 4.39504102677803828056437179469, 5.12640840160697463593928620698, 7.38599936240909339681364804143, 9.445081460010016499771947749226, 11.01706449775405089261266237962, 11.80240784080167058729983132416, 13.83550610160791396192164991618, 14.86965609544306747728503129161, 16.01286565797540793298010119901, 17.03307696654649896442688469490

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