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Properties

Label 2-26-13.9-c3-0-0
Degree $2$
Conductor $26$
Sign $-0.736 - 0.676i$
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 + (−2.93 + 5.07i)3-s + (−1.99 − 3.46i)4-s − 10.8·5-s + (−5.86 − 10.1i)6-s + (14.9 + 25.8i)7-s + 7.99·8-s + (−3.70 − 6.41i)9-s + (10.8 − 18.8i)10-s + (24.5 − 42.4i)11-s + 23.4·12-s + (2.29 + 46.8i)13-s − 59.7·14-s + (31.8 − 55.1i)15-s + (−8 + 13.8i)16-s + (3.03 + 5.26i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.564 + 0.977i)3-s + (−0.249 − 0.433i)4-s − 0.971·5-s + (−0.399 − 0.691i)6-s + (0.806 + 1.39i)7-s + 0.353·8-s + (−0.137 − 0.237i)9-s + (0.343 − 0.595i)10-s + (0.672 − 1.16i)11-s + 0.564·12-s + (0.0490 + 0.998i)13-s − 1.14·14-s + (0.548 − 0.950i)15-s + (−0.125 + 0.216i)16-s + (0.0433 + 0.0750i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $-0.736 - 0.676i$
Analytic conductor: \(1.53404\)
Root analytic conductor: \(1.23856\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{26} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 26,\ (\ :3/2),\ -0.736 - 0.676i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.262959 + 0.675205i\)
\(L(\frac12)\) \(\approx\) \(0.262959 + 0.675205i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.29 - 46.8i)T \)
good3 \( 1 + (2.93 - 5.07i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 10.8T + 125T^{2} \)
7 \( 1 + (-14.9 - 25.8i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-24.5 + 42.4i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-3.03 - 5.26i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-2.93 - 5.07i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-2.79 + 4.84i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-5.30 + 9.19i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 316.T + 2.97e4T^{2} \)
37 \( 1 + (-190. + 329. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (134. - 232. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-115. - 199. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 524.T + 1.03e5T^{2} \)
53 \( 1 + 274.T + 1.48e5T^{2} \)
59 \( 1 + (-71.4 - 123. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (281. + 487. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-258. + 448. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-72.2 - 125. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 201.T + 3.89e5T^{2} \)
79 \( 1 - 26.4T + 4.93e5T^{2} \)
83 \( 1 + 1.14e3T + 5.71e5T^{2} \)
89 \( 1 + (-331. + 574. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-68.7 - 119. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−17.10676345655043276596722203996, −16.06703343402039572046150297383, −15.41868663352088244978081586536, −14.24427792361761599816831430887, −11.82756938266003079774622420377, −11.13314163693425410830690907828, −9.267731101272801795526409829120, −8.142713511074802129203159405187, −5.95125914405951938939861489868, −4.42816908855874383243309172441, 0.982721169392091034320639105487, 4.25364232318150418633196518101, 7.08924542579971883408292112398, 7.952253335839071481181417898087, 10.22257445444040448223601338399, 11.53159958765513689109830589158, 12.33165535380828118137062802132, 13.65371640270481126152324795363, 15.28387857612509209683362483619, 17.21615574742369850724058810915

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