Properties

Label 2-26-13.3-c3-0-1
Degree $2$
Conductor $26$
Sign $0.477 - 0.878i$
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 + (1.5 + 2.59i)3-s + (−1.99 + 3.46i)4-s + 2·5-s + (−3 + 5.19i)6-s + (2.5 − 4.33i)7-s − 7.99·8-s + (9 − 15.5i)9-s + (2 + 3.46i)10-s + (−6.5 − 11.2i)11-s − 12·12-s + (−13 − 45.0i)13-s + 10·14-s + (3 + 5.19i)15-s + (−8 − 13.8i)16-s + (−13.5 + 23.3i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + 0.178·5-s + (−0.204 + 0.353i)6-s + (0.134 − 0.233i)7-s − 0.353·8-s + (0.333 − 0.577i)9-s + (0.0632 + 0.109i)10-s + (−0.178 − 0.308i)11-s − 0.288·12-s + (−0.277 − 0.960i)13-s + 0.190·14-s + (0.0516 + 0.0894i)15-s + (−0.125 − 0.216i)16-s + (−0.192 + 0.333i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $0.477 - 0.878i$
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: \((2,\ 26,\ (\ :3/2),\ 0.477 - 0.878i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.24307 + 0.739100i\)
\(L(\frac12)\) \(\approx\) \(1.24307 + 0.739100i\)
\(L(\frac{5}{2})\) 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 \( 1 + (13 + 45.0i)T \)
good3 \( 1 + (-1.5 - 2.59i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 2T + 125T^{2} \)
7 \( 1 + (-2.5 + 4.33i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (6.5 + 11.2i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (13.5 - 23.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (37.5 - 64.9i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-93.5 - 161. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-6.5 - 11.2i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 104T + 2.97e4T^{2} \)
37 \( 1 + (211.5 + 366. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (97.5 + 168. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (99.5 - 172. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 388T + 1.03e5T^{2} \)
53 \( 1 - 618T + 1.48e5T^{2} \)
59 \( 1 + (245.5 - 425. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (87.5 - 151. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (408.5 + 707. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (39.5 - 68.4i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 230T + 3.89e5T^{2} \)
79 \( 1 - 764T + 4.93e5T^{2} \)
83 \( 1 + 732T + 5.71e5T^{2} \)
89 \( 1 + (-520.5 - 901. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-48.5 + 84.0i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−17.06657722100360052393751721871, −15.67931106515282247309160684598, −14.93083331342688660038974078906, −13.63862169255085041401107406386, −12.37921456433539596559586339320, −10.50815696833532549658201112137, −9.071524386024832203409592637399, −7.49105662780509970496686114568, −5.64710886266161852550261434252, −3.73783482224117836944406712285, 2.22076700815509812169468933899, 4.75844192131633740885301717182, 6.92065287942029342012414399001, 8.768842301600484894537865311435, 10.34289712283183654728287907053, 11.78077604749722952198285088460, 13.03581064192076216499193145662, 13.96053838370182530914030829591, 15.27219232851747151098632200614, 16.86515815668010914274020262267

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