Properties

Label 2-13-13.2-c2-0-0
Degree $2$
Conductor $13$
Sign $0.999 + 0.0257i$
Analytic cond. $0.354224$
Root an. cond. $0.595167$
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  + (−0.5 − 0.133i)2-s + (0.366 + 0.633i)3-s + (−3.23 − 1.86i)4-s + (−2.63 + 2.63i)5-s + (−0.0980 − 0.366i)6-s + (5.73 − 1.53i)7-s + (2.83 + 2.83i)8-s + (4.23 − 7.33i)9-s + (1.66 − 0.964i)10-s + (−4.19 + 15.6i)11-s − 2.73i·12-s + (−6.5 − 11.2i)13-s − 3.07·14-s + (−2.63 − 0.705i)15-s + (6.42 + 11.1i)16-s + (−15.9 − 9.23i)17-s + ⋯
L(s)  = 1  + (−0.250 − 0.0669i)2-s + (0.122 + 0.211i)3-s + (−0.808 − 0.466i)4-s + (−0.526 + 0.526i)5-s + (−0.0163 − 0.0610i)6-s + (0.818 − 0.219i)7-s + (0.353 + 0.353i)8-s + (0.470 − 0.814i)9-s + (0.166 − 0.0964i)10-s + (−0.381 + 1.42i)11-s − 0.227i·12-s + (−0.5 − 0.866i)13-s − 0.219·14-s + (−0.175 − 0.0470i)15-s + (0.401 + 0.695i)16-s + (−0.940 − 0.543i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0257i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.999 + 0.0257i$
Analytic conductor: \(0.354224\)
Root analytic conductor: \(0.595167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :1),\ 0.999 + 0.0257i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.619000 - 0.00798248i\)
\(L(\frac12)\) \(\approx\) \(0.619000 - 0.00798248i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (6.5 + 11.2i)T \)
good2 \( 1 + (0.5 + 0.133i)T + (3.46 + 2i)T^{2} \)
3 \( 1 + (-0.366 - 0.633i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (2.63 - 2.63i)T - 25iT^{2} \)
7 \( 1 + (-5.73 + 1.53i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (4.19 - 15.6i)T + (-104. - 60.5i)T^{2} \)
17 \( 1 + (15.9 + 9.23i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-1.63 - 6.09i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (-17.4 + 10.0i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (4.69 + 8.13i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (11.9 - 11.9i)T - 961iT^{2} \)
37 \( 1 + (-8.11 + 30.2i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (-44.9 - 12.0i)T + (1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (-45 - 25.9i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (34.3 + 34.3i)T + 2.20e3iT^{2} \)
53 \( 1 + 14.7T + 2.80e3T^{2} \)
59 \( 1 + (92.9 - 24.9i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (12.8 - 22.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-39.0 - 10.4i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (11.9 + 44.6i)T + (-4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (19.2 + 19.2i)T + 5.32e3iT^{2} \)
79 \( 1 - 62.7T + 6.24e3T^{2} \)
83 \( 1 + (24.4 - 24.4i)T - 6.88e3iT^{2} \)
89 \( 1 + (23.1 - 86.4i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-14.1 - 52.9i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−19.76991606705878862998868980616, −18.21182317074686779649978089077, −17.64350259213821990500938318148, −15.29071013712631202961537634294, −14.60793667814246444314596565503, −12.73506263469685486997756218993, −10.80114878676248533836218564486, −9.441057435701829335531531376602, −7.51474711131236354201642582585, −4.61993023668330184227022435093, 4.65942459653037270782916514650, 7.82745848784865698386406467506, 8.882317382453934682412255272055, 11.17170965216970607950473066331, 12.87656638065065882475295154215, 14.02871999898276824260912970034, 15.99527459313062271554975282177, 17.13365449821973997528271361000, 18.56541507849067410742095645973, 19.46363117579604263318838230277

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