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Properties

Label 2-26-13.8-c2-0-0
Degree $2$
Conductor $26$
Sign $0.881 - 0.471i$
Analytic cond. $0.708448$
Root an. cond. $0.841693$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)2-s + 2i·4-s + (−3 − 3i)5-s + (2 − 2i)7-s + (−2 + 2i)8-s − 9·9-s − 6i·10-s + (6 − 6i)11-s + 13i·13-s + 4·14-s − 4·16-s + 6i·17-s + (−9 − 9i)18-s + (26 + 26i)19-s + (6 − 6i)20-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + 0.5i·4-s + (−0.600 − 0.600i)5-s + (0.285 − 0.285i)7-s + (−0.250 + 0.250i)8-s − 9-s − 0.600i·10-s + (0.545 − 0.545i)11-s + i·13-s + 0.285·14-s − 0.250·16-s + 0.352i·17-s + (−0.5 − 0.5i)18-s + (1.36 + 1.36i)19-s + (0.300 − 0.300i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $0.881 - 0.471i$
Analytic conductor: \(0.708448\)
Root analytic conductor: \(0.841693\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{26} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 26,\ (\ :1),\ 0.881 - 0.471i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.05219 + 0.263852i\)
\(L(\frac12)\) \(\approx\) \(1.05219 + 0.263852i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
13 \( 1 - 13iT \)
good3 \( 1 + 9T^{2} \)
5 \( 1 + (3 + 3i)T + 25iT^{2} \)
7 \( 1 + (-2 + 2i)T - 49iT^{2} \)
11 \( 1 + (-6 + 6i)T - 121iT^{2} \)
17 \( 1 - 6iT - 289T^{2} \)
19 \( 1 + (-26 - 26i)T + 361iT^{2} \)
23 \( 1 + 24iT - 529T^{2} \)
29 \( 1 + 48T + 841T^{2} \)
31 \( 1 + (14 + 14i)T + 961iT^{2} \)
37 \( 1 + (-37 + 37i)T - 1.36e3iT^{2} \)
41 \( 1 + (9 + 9i)T + 1.68e3iT^{2} \)
43 \( 1 - 36iT - 1.84e3T^{2} \)
47 \( 1 + (-42 + 42i)T - 2.20e3iT^{2} \)
53 \( 1 - 30T + 2.80e3T^{2} \)
59 \( 1 + (54 - 54i)T - 3.48e3iT^{2} \)
61 \( 1 + 18T + 3.72e3T^{2} \)
67 \( 1 + (22 + 22i)T + 4.48e3iT^{2} \)
71 \( 1 + (-6 - 6i)T + 5.04e3iT^{2} \)
73 \( 1 + (-17 + 17i)T - 5.32e3iT^{2} \)
79 \( 1 + 108T + 6.24e3T^{2} \)
83 \( 1 + (-78 - 78i)T + 6.88e3iT^{2} \)
89 \( 1 + (9 - 9i)T - 7.92e3iT^{2} \)
97 \( 1 + (47 + 47i)T + 9.40e3iT^{2} \)
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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

−16.75075000834519460076311928876, −16.44712681731140219534814499481, −14.70294232960147805991572306730, −13.90300393988905715008427178251, −12.27189113398734597220008821566, −11.30287356242241452065891629106, −9.001743277187219891262692603270, −7.74904870654413418110336879363, −5.84602179849635622379922007877, −4.00527522059654166627719770843, 3.18651302706180249264320587061, 5.38569459593678042418592606655, 7.41588762584680208753980303552, 9.331167023136773648796773316813, 11.12671721156407251343402286710, 11.80008865402941539056754557416, 13.41921440535433854909104938686, 14.72775324191117174237516499556, 15.50405230739533236134390730981, 17.35449125857574115996580014573

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