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Properties

Label 2-26-13.4-c3-0-3
Degree $2$
Conductor $26$
Sign $0.155 + 0.987i$
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 + (1.47 − 2.56i)3-s + (1.99 + 3.46i)4-s − 20.2i·5-s + (−5.12 + 2.95i)6-s + (−1.04 + 0.603i)7-s − 7.99i·8-s + (9.13 + 15.8i)9-s + (−20.2 + 35.0i)10-s + (35.3 + 20.4i)11-s + 11.8·12-s + (−36.8 + 29.0i)13-s + 2.41·14-s + (−51.7 − 29.9i)15-s + (−8 + 13.8i)16-s + (4.90 + 8.49i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.284 − 0.492i)3-s + (0.249 + 0.433i)4-s − 1.80i·5-s + (−0.348 + 0.201i)6-s + (−0.0564 + 0.0326i)7-s − 0.353i·8-s + (0.338 + 0.585i)9-s + (−0.639 + 1.10i)10-s + (0.969 + 0.559i)11-s + 0.284·12-s + (−0.785 + 0.618i)13-s + 0.0461·14-s + (−0.891 − 0.514i)15-s + (−0.125 + 0.216i)16-s + (0.0699 + 0.121i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $0.155 + 0.987i$
Analytic conductor: \(1.53404\)
Root analytic conductor: \(1.23856\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{26} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 26,\ (\ :3/2),\ 0.155 + 0.987i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.740469 - 0.632717i\)
\(L(\frac12)\) \(\approx\) \(0.740469 - 0.632717i\)
\(L(\frac{5}{2})\) 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 \( 1 + (36.8 - 29.0i)T \)
good3 \( 1 + (-1.47 + 2.56i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 20.2iT - 125T^{2} \)
7 \( 1 + (1.04 - 0.603i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-35.3 - 20.4i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-4.90 - 8.49i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-98.7 + 57.0i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (11.6 - 20.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-31.7 + 55.0i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 225. iT - 2.97e4T^{2} \)
37 \( 1 + (154. + 89.2i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (70.6 + 40.7i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-164. - 284. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 560. iT - 1.03e5T^{2} \)
53 \( 1 + 389.T + 1.48e5T^{2} \)
59 \( 1 + (189. - 109. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-180. - 311. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (475. + 274. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (689. - 397. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 140. iT - 3.89e5T^{2} \)
79 \( 1 - 568.T + 4.93e5T^{2} \)
83 \( 1 - 1.09e3iT - 5.71e5T^{2} \)
89 \( 1 + (760. + 438. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-1.37e3 + 794. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−16.82869838399006707348411536987, −15.92486906392024615489433126240, −13.88148675608352281793243595311, −12.64761183126550726043814731781, −11.85844185208383344165663643858, −9.694689234930773539898081948233, −8.735416058314895116692319527460, −7.32897958601201697832511984849, −4.73283295372721493181530228728, −1.49915896595163907660851419763, 3.30659998857122552043519384155, 6.29622170845039603083225160867, 7.53584774224580991021563361460, 9.498325387408235420420355844433, 10.40965042958356472520669941294, 11.75824269503622841056602178866, 14.17864357073559192369067309043, 14.81576164919266349511202079888, 15.85968883597927490997689957648, 17.40827400669621164589412251854

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