Properties

Label 2-65-13.10-c1-0-1
Degree $2$
Conductor $65$
Sign $0.651 - 0.758i$
Analytic cond. $0.519027$
Root an. cond. $0.720435$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.190 + 0.109i)2-s + (0.800 + 1.38i)3-s + (−0.975 + 1.69i)4-s i·5-s + (−0.304 − 0.175i)6-s + (−0.287 − 0.166i)7-s − 0.868i·8-s + (0.219 − 0.380i)9-s + (0.109 + 0.190i)10-s + (4.65 − 2.68i)11-s − 3.12·12-s + (−3.55 − 0.619i)13-s + 0.0729·14-s + (1.38 − 0.800i)15-s + (−1.85 − 3.21i)16-s + (−2.53 + 4.38i)17-s + ⋯
L(s)  = 1  + (−0.134 + 0.0776i)2-s + (0.461 + 0.800i)3-s + (−0.487 + 0.845i)4-s − 0.447i·5-s + (−0.124 − 0.0717i)6-s + (−0.108 − 0.0627i)7-s − 0.306i·8-s + (0.0732 − 0.126i)9-s + (0.0347 + 0.0601i)10-s + (1.40 − 0.809i)11-s − 0.901·12-s + (−0.985 − 0.171i)13-s + 0.0195·14-s + (0.357 − 0.206i)15-s + (−0.464 − 0.803i)16-s + (−0.614 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.651 - 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.651 - 0.758i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.651 - 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.801290 + 0.368313i\)
\(L(\frac12)\) \(\approx\) \(0.801290 + 0.368313i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + iT \)
13 \( 1 + (3.55 + 0.619i)T \)
good2 \( 1 + (0.190 - 0.109i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.800 - 1.38i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.287 + 0.166i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.65 + 2.68i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.53 - 4.38i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.96 + 1.13i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.41 + 2.45i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.45 - 2.51i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.46iT - 31T^{2} \)
37 \( 1 + (5.17 - 2.98i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.23 + 1.86i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.53 - 4.38i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.34iT - 47T^{2} \)
53 \( 1 + 1.56T + 53T^{2} \)
59 \( 1 + (-2.34 - 1.35i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.05 - 12.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.94 + 5.16i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (11.0 + 6.39i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.68iT - 73T^{2} \)
79 \( 1 - 4.51T + 79T^{2} \)
83 \( 1 + 4.26iT - 83T^{2} \)
89 \( 1 + (2.79 - 1.61i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.17 - 1.25i)T + (48.5 + 84.0i)T^{2} \)
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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

−15.03475765989790546639160388568, −14.08034281759538355325928148957, −12.80949798648276166353144607573, −11.90231065390357000809432293997, −10.23487566878325788359240402754, −9.018755820104747997268103713093, −8.510132644312566629443595730688, −6.71788102904751183275109623967, −4.55811518716557162412556913308, −3.51165273711499983201429723159, 2.03773300000244614306454951812, 4.56804986037642860765581972549, 6.43985310246550613953770658146, 7.50198416305323760691918524689, 9.126067418043696733670669210564, 9.958380409594795905148953487384, 11.44146993080559789274857062193, 12.65687830642304021895992346384, 13.93914012754837913755165049793, 14.41937148184295279441568938881

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