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Properties

Label 2-65-65.33-c1-0-2
Degree $2$
Conductor $65$
Sign $0.880 - 0.473i$
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  + (−0.246 + 0.427i)2-s + (0.908 − 0.243i)3-s + (0.878 + 1.52i)4-s + (−2.21 − 0.284i)5-s + (−0.120 + 0.448i)6-s + (3.18 − 1.83i)7-s − 1.85·8-s + (−1.83 + 1.05i)9-s + (0.669 − 0.878i)10-s + (−0.177 − 0.664i)11-s + (1.16 + 1.16i)12-s + (−2.11 − 2.92i)13-s + 1.81i·14-s + (−2.08 + 0.281i)15-s + (−1.29 + 2.24i)16-s + (0.614 − 2.29i)17-s + ⋯
L(s)  = 1  + (−0.174 + 0.302i)2-s + (0.524 − 0.140i)3-s + (0.439 + 0.760i)4-s + (−0.991 − 0.127i)5-s + (−0.0490 + 0.183i)6-s + (1.20 − 0.694i)7-s − 0.655·8-s + (−0.610 + 0.352i)9-s + (0.211 − 0.277i)10-s + (−0.0536 − 0.200i)11-s + (0.337 + 0.337i)12-s + (−0.585 − 0.810i)13-s + 0.485i·14-s + (−0.538 + 0.0726i)15-s + (−0.324 + 0.562i)16-s + (0.149 − 0.556i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.880 - 0.473i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.880 - 0.473i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.897123 + 0.226011i\)
\(L(\frac12)\) \(\approx\) \(0.897123 + 0.226011i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.21 + 0.284i)T \)
13 \( 1 + (2.11 + 2.92i)T \)
good2 \( 1 + (0.246 - 0.427i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.908 + 0.243i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-3.18 + 1.83i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.177 + 0.664i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.614 + 2.29i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (5.29 + 1.41i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.350 - 1.30i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-8.24 - 4.75i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.81 - 4.81i)T + 31iT^{2} \)
37 \( 1 + (-1.58 - 0.917i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.534 - 0.143i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.09 + 0.560i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 3.80iT - 47T^{2} \)
53 \( 1 + (2.47 + 2.47i)T + 53iT^{2} \)
59 \( 1 + (-2.69 + 10.0i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.09 + 5.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.12 - 10.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.73 - 6.47i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 3.37T + 73T^{2} \)
79 \( 1 + 3.12iT - 79T^{2} \)
83 \( 1 - 2.13iT - 83T^{2} \)
89 \( 1 + (3.26 - 0.874i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (3.53 + 6.12i)T + (-48.5 + 84.0i)T^{2} \)
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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

−15.01150761132530540774618215617, −14.10833813186078749140818033032, −12.70838199458803559724053871314, −11.61273111058021635364077530301, −10.74991829429104471606032401590, −8.488967541040984759467140388047, −8.076908737128926219749758483154, −7.05840126408489486051835138837, −4.74270536652492058106426502831, −3.00707812523460288609840140663, 2.40166957102956391690475466466, 4.52275232332605578732314684377, 6.28130803462563181164598894132, 8.012570892265524227362598958956, 8.915439261738999416456203151506, 10.39600903827274445130703714180, 11.59882044104803595031452808671, 12.05925584548486967272737009793, 14.20314383691893749855679665616, 14.96123380725928700734064091628

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