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Properties

Label 2-26-13.10-c3-0-2
Degree $2$
Conductor $26$
Sign $-0.00872 + 0.999i$
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 + (−2.97 − 5.15i)3-s + (1.99 − 3.46i)4-s − 13.0i·5-s + (10.3 + 5.95i)6-s + (−3.11 − 1.79i)7-s + 7.99i·8-s + (−4.23 + 7.34i)9-s + (13.0 + 22.5i)10-s + (−24.2 + 14.0i)11-s − 23.8·12-s + (40.6 + 23.2i)13-s + 7.19·14-s + (−67.2 + 38.8i)15-s + (−8 − 13.8i)16-s + (57.7 − 100. i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.573 − 0.992i)3-s + (0.249 − 0.433i)4-s − 1.16i·5-s + (0.701 + 0.405i)6-s + (−0.168 − 0.0970i)7-s + 0.353i·8-s + (−0.156 + 0.271i)9-s + (0.412 + 0.713i)10-s + (−0.665 + 0.384i)11-s − 0.573·12-s + (0.868 + 0.496i)13-s + 0.137·14-s + (−1.15 + 0.668i)15-s + (−0.125 − 0.216i)16-s + (0.823 − 1.42i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00872 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.00872 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $-0.00872 + 0.999i$
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: \((2,\ 26,\ (\ :3/2),\ -0.00872 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.496674 - 0.501025i\)
\(L(\frac12)\) \(\approx\) \(0.496674 - 0.501025i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-40.6 - 23.2i)T \)
good3 \( 1 + (2.97 + 5.15i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 13.0iT - 125T^{2} \)
7 \( 1 + (3.11 + 1.79i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (24.2 - 14.0i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-57.7 + 100. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-112. - 64.7i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (87.4 + 151. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-109. - 189. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 53.3iT - 2.97e4T^{2} \)
37 \( 1 + (-109. + 63.3i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (222. - 128. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (14.2 - 24.6i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 108. iT - 1.03e5T^{2} \)
53 \( 1 - 215.T + 1.48e5T^{2} \)
59 \( 1 + (-339. - 196. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-112. + 194. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (120. - 69.7i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-550. - 317. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 946. iT - 3.89e5T^{2} \)
79 \( 1 + 198.T + 4.93e5T^{2} \)
83 \( 1 - 692. iT - 5.71e5T^{2} \)
89 \( 1 + (701. - 405. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (996. + 575. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−16.55217940869456619879557918407, −16.11006269898289355408949764971, −14.01040961569509531698060595997, −12.68205948796474674372093533117, −11.75500982744109767579485711955, −9.845086754049143981238163275861, −8.328890091147746845120301271028, −6.95967192307855204579737419440, −5.34805856046314187243021651086, −1.01586042840148154240267235635, 3.40771921721703011462717059860, 5.86309079276432383201442935231, 7.85356776112401527960898669323, 9.851061972931510924248711029377, 10.62830855168434337398105627642, 11.56765774818054577882321110579, 13.52125328349379869361984674038, 15.30540520781556221976953112565, 15.99902211923655126458467074330, 17.39007303685020502064267282377

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