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Properties

Label 2-26-13.8-c4-0-4
Degree $2$
Conductor $26$
Sign $0.887 - 0.460i$
Analytic cond. $2.68761$
Root an. cond. $1.63939$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 + 2i)2-s + 15.6·3-s + 8i·4-s + (−24.8 − 24.8i)5-s + (31.3 + 31.3i)6-s + (−19.1 + 19.1i)7-s + (−16 + 16i)8-s + 165.·9-s − 99.4i·10-s + (−121. + 121. i)11-s + 125. i·12-s + (2.23 − 168. i)13-s − 76.6·14-s + (−389. − 389. i)15-s − 64·16-s − 214. i·17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + 1.74·3-s + 0.5i·4-s + (−0.994 − 0.994i)5-s + (0.871 + 0.871i)6-s + (−0.391 + 0.391i)7-s + (−0.250 + 0.250i)8-s + 2.03·9-s − 0.994i·10-s + (−1.00 + 1.00i)11-s + 0.871i·12-s + (0.0132 − 0.999i)13-s − 0.391·14-s + (−1.73 − 1.73i)15-s − 0.250·16-s − 0.740i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $0.887 - 0.460i$
Analytic conductor: \(2.68761\)
Root analytic conductor: \(1.63939\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{26} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 26,\ (\ :2),\ 0.887 - 0.460i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.18123 + 0.531696i\)
\(L(\frac12)\) \(\approx\) \(2.18123 + 0.531696i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 - 2i)T \)
13 \( 1 + (-2.23 + 168. i)T \)
good3 \( 1 - 15.6T + 81T^{2} \)
5 \( 1 + (24.8 + 24.8i)T + 625iT^{2} \)
7 \( 1 + (19.1 - 19.1i)T - 2.40e3iT^{2} \)
11 \( 1 + (121. - 121. i)T - 1.46e4iT^{2} \)
17 \( 1 + 214. iT - 8.35e4T^{2} \)
19 \( 1 + (-261. - 261. i)T + 1.30e5iT^{2} \)
23 \( 1 + 200. iT - 2.79e5T^{2} \)
29 \( 1 - 384.T + 7.07e5T^{2} \)
31 \( 1 + (-525. - 525. i)T + 9.23e5iT^{2} \)
37 \( 1 + (-553. + 553. i)T - 1.87e6iT^{2} \)
41 \( 1 + (495. + 495. i)T + 2.82e6iT^{2} \)
43 \( 1 - 2.76e3iT - 3.41e6T^{2} \)
47 \( 1 + (861. - 861. i)T - 4.87e6iT^{2} \)
53 \( 1 + 3.72e3T + 7.89e6T^{2} \)
59 \( 1 + (-3.76e3 + 3.76e3i)T - 1.21e7iT^{2} \)
61 \( 1 - 2.52e3T + 1.38e7T^{2} \)
67 \( 1 + (1.77e3 + 1.77e3i)T + 2.01e7iT^{2} \)
71 \( 1 + (2.83e3 + 2.83e3i)T + 2.54e7iT^{2} \)
73 \( 1 + (-4.98e3 + 4.98e3i)T - 2.83e7iT^{2} \)
79 \( 1 + 6.67e3T + 3.89e7T^{2} \)
83 \( 1 + (3.47e3 + 3.47e3i)T + 4.74e7iT^{2} \)
89 \( 1 + (4.34e3 - 4.34e3i)T - 6.27e7iT^{2} \)
97 \( 1 + (5.98e3 + 5.98e3i)T + 8.85e7iT^{2} \)
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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

−15.96307049675433631126287998348, −15.62071607245575269515220939878, −14.44863977183881419469631514310, −13.04868880268756543868585065678, −12.39315786012748738975942963889, −9.682411328022075057294577542794, −8.285475580027662213207307587328, −7.60926875620586321041122124475, −4.73010947629280020763726814563, −3.04348057787614206528864373222, 2.82780173502235486113228223383, 3.84706196363591393462274058329, 7.08935714643347299642031358753, 8.408495914497962438959803375887, 10.05033283947140738564187437209, 11.40565786372837007688826693844, 13.25615878473293523798767071702, 14.01003844302389353595123311630, 15.10462588286895378809671697809, 15.93322181808674604144881761119

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