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Properties

Label 2-26-13.10-c3-0-1
Degree $2$
Conductor $26$
Sign $0.999 + 0.0174i$
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 + (2.43 + 4.22i)3-s + (1.99 − 3.46i)4-s − 0.110i·5-s + (8.44 + 4.87i)6-s + (−14.0 − 8.09i)7-s − 7.99i·8-s + (1.61 − 2.79i)9-s + (−0.110 − 0.190i)10-s + (−33.6 + 19.4i)11-s + 19.5·12-s + (−23.6 + 40.4i)13-s − 32.3·14-s + (0.464 − 0.268i)15-s + (−8 − 13.8i)16-s + (47.2 − 81.7i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.469 + 0.812i)3-s + (0.249 − 0.433i)4-s − 0.00985i·5-s + (0.574 + 0.331i)6-s + (−0.757 − 0.437i)7-s − 0.353i·8-s + (0.0598 − 0.103i)9-s + (−0.00348 − 0.00603i)10-s + (−0.921 + 0.531i)11-s + 0.469·12-s + (−0.503 + 0.863i)13-s − 0.618·14-s + (0.00800 − 0.00462i)15-s + (−0.125 − 0.216i)16-s + (0.673 − 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0174i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $0.999 + 0.0174i$
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: \((2,\ 26,\ (\ :3/2),\ 0.999 + 0.0174i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.63283 - 0.0142410i\)
\(L(\frac12)\) \(\approx\) \(1.63283 - 0.0142410i\)
\(L(\frac{5}{2})\) 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 \( 1 + (23.6 - 40.4i)T \)
good3 \( 1 + (-2.43 - 4.22i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 0.110iT - 125T^{2} \)
7 \( 1 + (14.0 + 8.09i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (33.6 - 19.4i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-47.2 + 81.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-33.5 - 19.3i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-46.2 - 80.0i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (96.2 + 166. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 158. iT - 2.97e4T^{2} \)
37 \( 1 + (248. - 143. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-202. + 117. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (262. - 455. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 320. iT - 1.03e5T^{2} \)
53 \( 1 - 414.T + 1.48e5T^{2} \)
59 \( 1 + (223. + 128. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (71.6 - 124. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-392. + 226. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (654. + 377. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 641. iT - 3.89e5T^{2} \)
79 \( 1 + 588.T + 4.93e5T^{2} \)
83 \( 1 + 744. iT - 5.71e5T^{2} \)
89 \( 1 + (1.08e3 - 624. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-1.04e3 - 602. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−16.54104398882911042154531093508, −15.61166156352329824956436464759, −14.48429561989058178593993479889, −13.29942048445696098724489792567, −11.95857993821099167647606032529, −10.24060623525230718543133550891, −9.429542892189717421987961931420, −7.09635475575726851359032403535, −4.87475004244263552545823236634, −3.22356954145946408930322157494, 2.90365294296524415459433277778, 5.55468829146875330471866140652, 7.21513721748874776851318303973, 8.463191378211974595111351416267, 10.53272498443640149045796179555, 12.61624230587131364636331151541, 12.99563135802997694282216700328, 14.40004064718983172853667824856, 15.60151473060152770783554437572, 16.77569600539440384168268980546

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