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Properties

Label 2-65-65.7-c1-0-0
Degree $2$
Conductor $65$
Sign $0.899 + 0.436i$
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  + (−2.29 − 1.32i)2-s + (0.335 + 1.25i)3-s + (2.51 + 4.34i)4-s + (1.81 − 1.30i)5-s + (0.889 − 3.31i)6-s + (0.0561 + 0.0972i)7-s − 8.00i·8-s + (1.14 − 0.658i)9-s + (−5.89 + 0.585i)10-s + (0.479 + 1.78i)11-s + (−4.60 + 4.60i)12-s + (2.96 + 2.05i)13-s − 0.297i·14-s + (2.24 + 1.83i)15-s + (−5.58 + 9.67i)16-s + (−2.63 − 0.706i)17-s + ⋯
L(s)  = 1  + (−1.62 − 0.936i)2-s + (0.193 + 0.723i)3-s + (1.25 + 2.17i)4-s + (0.812 − 0.583i)5-s + (0.363 − 1.35i)6-s + (0.0212 + 0.0367i)7-s − 2.83i·8-s + (0.380 − 0.219i)9-s + (−1.86 + 0.185i)10-s + (0.144 + 0.539i)11-s + (−1.32 + 1.32i)12-s + (0.821 + 0.569i)13-s − 0.0795i·14-s + (0.579 + 0.474i)15-s + (−1.39 + 2.41i)16-s + (−0.639 − 0.171i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.899 + 0.436i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.899 + 0.436i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.530136 - 0.121786i\)
\(L(\frac12)\) \(\approx\) \(0.530136 - 0.121786i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.81 + 1.30i)T \)
13 \( 1 + (-2.96 - 2.05i)T \)
good2 \( 1 + (2.29 + 1.32i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.335 - 1.25i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.0561 - 0.0972i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.479 - 1.78i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.63 + 0.706i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (6.72 + 1.80i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.10 - 0.831i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (4.03 + 2.32i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.624 + 0.624i)T + 31iT^{2} \)
37 \( 1 + (0.737 - 1.27i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.24 - 1.40i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.00 + 3.76i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 0.345T + 47T^{2} \)
53 \( 1 + (3.59 - 3.59i)T - 53iT^{2} \)
59 \( 1 + (-0.332 + 1.24i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.39 - 2.41i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.124 - 0.0721i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.41 - 5.28i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + 9.06iT - 73T^{2} \)
79 \( 1 - 15.1iT - 79T^{2} \)
83 \( 1 - 8.53T + 83T^{2} \)
89 \( 1 + (-0.549 + 0.147i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (12.9 - 7.48i)T + (48.5 - 84.0i)T^{2} \)
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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

−15.25347784428511995515023134164, −13.35053210918797641105644261217, −12.29632636388753150480503305262, −10.99609880138156020422542647807, −10.04136413872545063953351441038, −9.237266312586820706282399344000, −8.509926560755307946474527908544, −6.71476583432797695444819256843, −4.12464858090498011693084868742, −1.95386908057509669650738742730, 1.84899246907087217180698344704, 5.97669797018592171287519094517, 6.73600275697428691814029130452, 7.964370285043142306459367538820, 8.913035005532803550309216101705, 10.28172979931931842211740409745, 10.93942834701055851685431621684, 13.04099677985943369514553278963, 14.17524618145366206946797435765, 15.21154888652193364865306755191

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