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Properties

Label 2-13-13.5-c4-0-2
Degree $2$
Conductor $13$
Sign $0.706 + 0.707i$
Analytic cond. $1.34380$
Root an. cond. $1.15922$
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  + (3.48 − 3.48i)2-s − 1.36·3-s − 8.33i·4-s + (−6.84 + 6.84i)5-s + (−4.74 + 4.74i)6-s + (15.2 + 15.2i)7-s + (26.7 + 26.7i)8-s − 79.1·9-s + 47.7i·10-s + (−94.2 − 94.2i)11-s + 11.3i·12-s + (149. − 79.6i)13-s + 106.·14-s + (9.31 − 9.31i)15-s + 319.·16-s − 349. i·17-s + ⋯
L(s)  = 1  + (0.872 − 0.872i)2-s − 0.151·3-s − 0.521i·4-s + (−0.273 + 0.273i)5-s + (−0.131 + 0.131i)6-s + (0.312 + 0.312i)7-s + (0.417 + 0.417i)8-s − 0.977·9-s + 0.477i·10-s + (−0.779 − 0.779i)11-s + 0.0787i·12-s + (0.881 − 0.471i)13-s + 0.544·14-s + (0.0413 − 0.0413i)15-s + 1.24·16-s − 1.21i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.706 + 0.707i$
Analytic conductor: \(1.34380\)
Root analytic conductor: \(1.15922\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :2),\ 0.706 + 0.707i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.39183 - 0.576940i\)
\(L(\frac12)\) \(\approx\) \(1.39183 - 0.576940i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-149. + 79.6i)T \)
good2 \( 1 + (-3.48 + 3.48i)T - 16iT^{2} \)
3 \( 1 + 1.36T + 81T^{2} \)
5 \( 1 + (6.84 - 6.84i)T - 625iT^{2} \)
7 \( 1 + (-15.2 - 15.2i)T + 2.40e3iT^{2} \)
11 \( 1 + (94.2 + 94.2i)T + 1.46e4iT^{2} \)
17 \( 1 + 349. iT - 8.35e4T^{2} \)
19 \( 1 + (217. - 217. i)T - 1.30e5iT^{2} \)
23 \( 1 - 310. iT - 2.79e5T^{2} \)
29 \( 1 - 1.07e3T + 7.07e5T^{2} \)
31 \( 1 + (334. - 334. i)T - 9.23e5iT^{2} \)
37 \( 1 + (458. + 458. i)T + 1.87e6iT^{2} \)
41 \( 1 + (-1.40e3 + 1.40e3i)T - 2.82e6iT^{2} \)
43 \( 1 - 3.17e3iT - 3.41e6T^{2} \)
47 \( 1 + (2.45e3 + 2.45e3i)T + 4.87e6iT^{2} \)
53 \( 1 + 2.63e3T + 7.89e6T^{2} \)
59 \( 1 + (190. + 190. i)T + 1.21e7iT^{2} \)
61 \( 1 - 3.51e3T + 1.38e7T^{2} \)
67 \( 1 + (2.01e3 - 2.01e3i)T - 2.01e7iT^{2} \)
71 \( 1 + (5.58e3 - 5.58e3i)T - 2.54e7iT^{2} \)
73 \( 1 + (-328. - 328. i)T + 2.83e7iT^{2} \)
79 \( 1 - 4.04e3T + 3.89e7T^{2} \)
83 \( 1 + (-5.52e3 + 5.52e3i)T - 4.74e7iT^{2} \)
89 \( 1 + (5.86e3 + 5.86e3i)T + 6.27e7iT^{2} \)
97 \( 1 + (1.08e4 - 1.08e4i)T - 8.85e7iT^{2} \)
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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

−19.31606574294199403570055137809, −17.85431694910144536505541721963, −16.13994550491211320831788463331, −14.42848898387259639125017106533, −13.25732879435191646491935616704, −11.70178008328772317257199636917, −10.84941116658297932101209745926, −8.255486743805247326931621823863, −5.44033687431510483445844758930, −3.10968951041503230963672299874, 4.55405711746080778770261517914, 6.30947448871213233121168780664, 8.213166794039858770941057051581, 10.70743721140277450898310539819, 12.61032656258559414449216799779, 13.97263892290221199972459466319, 15.14754917771465868682576543085, 16.33664192662461295084809404182, 17.61546089797781883735807660434, 19.44555594942740994069894408209

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