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Properties

Label 2-26-13.12-c3-0-1
Degree $2$
Conductor $26$
Sign $0.668 - 0.743i$
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  + 2i·2-s + 5.86·3-s − 4·4-s + 2.13i·5-s + 11.7i·6-s + 3.86i·7-s − 8i·8-s + 7.40·9-s − 4.26·10-s − 53.1i·11-s − 23.4·12-s + (−34.8 − 31.3i)13-s − 7.73·14-s + 12.5i·15-s + 16·16-s + 43.3·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.12·3-s − 0.5·4-s + 0.190i·5-s + 0.798i·6-s + 0.208i·7-s − 0.353i·8-s + 0.274·9-s − 0.135·10-s − 1.45i·11-s − 0.564·12-s + (−0.743 − 0.668i)13-s − 0.147·14-s + 0.215i·15-s + 0.250·16-s + 0.618·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $0.668 - 0.743i$
Analytic conductor: \(1.53404\)
Root analytic conductor: \(1.23856\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{26} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 26,\ (\ :3/2),\ 0.668 - 0.743i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.30676 + 0.582625i\)
\(L(\frac12)\) \(\approx\) \(1.30676 + 0.582625i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
13 \( 1 + (34.8 + 31.3i)T \)
good3 \( 1 - 5.86T + 27T^{2} \)
5 \( 1 - 2.13iT - 125T^{2} \)
7 \( 1 - 3.86iT - 343T^{2} \)
11 \( 1 + 53.1iT - 1.33e3T^{2} \)
17 \( 1 - 43.3T + 4.91e3T^{2} \)
19 \( 1 - 148. iT - 6.85e3T^{2} \)
23 \( 1 + 122.T + 1.21e4T^{2} \)
29 \( 1 - 83.8T + 2.43e4T^{2} \)
31 \( 1 - 190. iT - 2.97e4T^{2} \)
37 \( 1 - 131. iT - 5.06e4T^{2} \)
41 \( 1 + 387. iT - 6.89e4T^{2} \)
43 \( 1 - 74.6T + 7.95e4T^{2} \)
47 \( 1 - 298. iT - 1.03e5T^{2} \)
53 \( 1 - 100.T + 1.48e5T^{2} \)
59 \( 1 + 479. iT - 2.05e5T^{2} \)
61 \( 1 + 479.T + 2.26e5T^{2} \)
67 \( 1 + 415. iT - 3.00e5T^{2} \)
71 \( 1 - 293. iT - 3.57e5T^{2} \)
73 \( 1 + 106. iT - 3.89e5T^{2} \)
79 \( 1 + 906.T + 4.93e5T^{2} \)
83 \( 1 + 22.1iT - 5.71e5T^{2} \)
89 \( 1 + 665. iT - 7.04e5T^{2} \)
97 \( 1 - 1.25e3iT - 9.12e5T^{2} \)
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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

−16.92176636223801254712997046559, −15.78743064530436334166685922035, −14.43546864179543719204204547509, −14.00481764846978536474312589562, −12.35476467145229940344771952496, −10.25416713868834642465213254963, −8.692931887643468490743879765670, −7.81296976553339410365603752850, −5.80248490376858168948194418521, −3.29985472209482248535310188747, 2.44051449285608333447793341671, 4.47956431934978548046794255437, 7.41660573536672553267243002526, 8.992869145409516300863507791327, 9.975975782204954507491283327790, 11.79842336534639304178314011950, 13.06685637815673392802895377831, 14.24186943073609793244562963326, 15.18663735721497807442599027573, 17.00219218943995390495959815588

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