↑

Properties

Label 2-13-13.8-c2-0-0
Degree $2$
Conductor $13$
Sign $0.984 - 0.176i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.581 + 0.581i)2-s − 4.16·3-s − 3.32i·4-s + (3.58 + 3.58i)5-s + (−2.41 − 2.41i)6-s + (−4.58 + 4.58i)7-s + (4.25 − 4.25i)8-s + 8.32·9-s + 4.16i·10-s + (5.32 − 5.32i)11-s + 13.8i·12-s + (−5.90 − 11.5i)13-s − 5.32·14-s + (−14.9 − 14.9i)15-s − 8.35·16-s + 21.9i·17-s + ⋯
L(s)  = 1  + (0.290 + 0.290i)2-s − 1.38·3-s − 0.831i·4-s + (0.716 + 0.716i)5-s + (−0.403 − 0.403i)6-s + (−0.654 + 0.654i)7-s + (0.532 − 0.532i)8-s + 0.924·9-s + 0.416i·10-s + (0.484 − 0.484i)11-s + 1.15i·12-s + (−0.454 − 0.890i)13-s − 0.380·14-s + (−0.993 − 0.993i)15-s − 0.521·16-s + 1.29i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.984 - 0.176i$
Analytic conductor: \(0.354224\)
Root analytic conductor: \(0.595167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :1),\ 0.984 - 0.176i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.648809 + 0.0577555i\)
\(L(\frac12)\) \(\approx\) \(0.648809 + 0.0577555i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (5.90 + 11.5i)T \)
good2 \( 1 + (-0.581 - 0.581i)T + 4iT^{2} \)
3 \( 1 + 4.16T + 9T^{2} \)
5 \( 1 + (-3.58 - 3.58i)T + 25iT^{2} \)
7 \( 1 + (4.58 - 4.58i)T - 49iT^{2} \)
11 \( 1 + (-5.32 + 5.32i)T - 121iT^{2} \)
17 \( 1 - 21.9iT - 289T^{2} \)
19 \( 1 + (-3.16 - 3.16i)T + 361iT^{2} \)
23 \( 1 - 8.51iT - 529T^{2} \)
29 \( 1 + 5.81T + 841T^{2} \)
31 \( 1 + (-0.513 - 0.513i)T + 961iT^{2} \)
37 \( 1 + (-24.2 + 24.2i)T - 1.36e3iT^{2} \)
41 \( 1 + (-4.83 - 4.83i)T + 1.68e3iT^{2} \)
43 \( 1 + 30.4iT - 1.84e3T^{2} \)
47 \( 1 + (37.3 - 37.3i)T - 2.20e3iT^{2} \)
53 \( 1 + 35.8T + 2.80e3T^{2} \)
59 \( 1 + (-58.2 + 58.2i)T - 3.48e3iT^{2} \)
61 \( 1 + 80.3T + 3.72e3T^{2} \)
67 \( 1 + (-39.0 - 39.0i)T + 4.48e3iT^{2} \)
71 \( 1 + (-91.5 - 91.5i)T + 5.04e3iT^{2} \)
73 \( 1 + (-31.6 + 31.6i)T - 5.32e3iT^{2} \)
79 \( 1 + 18.7T + 6.24e3T^{2} \)
83 \( 1 + (44.6 + 44.6i)T + 6.88e3iT^{2} \)
89 \( 1 + (-8.89 + 8.89i)T - 7.92e3iT^{2} \)
97 \( 1 + (121. + 121. i)T + 9.40e3iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.49695197618202841717339595129, −18.33323662465042175742064478334, −17.18608481629438217751192576964, −15.76229596423075253929326431430, −14.45109489363855657383888504907, −12.73477724760375199072324025869, −10.99395777799112031275258853299, −9.896974242426991948391952809843, −6.38261627920717730506117029383, −5.65990377670370487912622172668, 4.77450417013775926355845063463, 6.85874272103586218025205582972, 9.563996724071608323299871578343, 11.43954033197981425354098322200, 12.48767843173433937554344724520, 13.67826964445310405211522983516, 16.51199170187262004701105122947, 16.81981330681481747967995918196, 17.95968323919624174885987371043, 20.07705563370172261207446866316

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