Properties

Label 2-65-65.28-c1-0-3
Degree $2$
Conductor $65$
Sign $0.810 + 0.585i$
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  + (−0.237 + 0.137i)2-s + (0.611 − 2.28i)3-s + (−0.962 + 1.66i)4-s + (1.45 − 1.69i)5-s + (0.168 + 0.627i)6-s + (−0.193 + 0.334i)7-s − 1.07i·8-s + (−2.23 − 1.29i)9-s + (−0.112 + 0.604i)10-s + (−1.12 + 4.21i)11-s + (3.21 + 3.21i)12-s + (−1.35 + 3.34i)13-s − 0.106i·14-s + (−2.98 − 4.35i)15-s + (−1.77 − 3.07i)16-s + (1.90 − 0.510i)17-s + ⋯
L(s)  = 1  + (−0.168 + 0.0971i)2-s + (0.353 − 1.31i)3-s + (−0.481 + 0.833i)4-s + (0.650 − 0.759i)5-s + (0.0686 + 0.256i)6-s + (−0.0729 + 0.126i)7-s − 0.381i·8-s + (−0.745 − 0.430i)9-s + (−0.0356 + 0.191i)10-s + (−0.340 + 1.27i)11-s + (0.928 + 0.928i)12-s + (−0.376 + 0.926i)13-s − 0.0283i·14-s + (−0.771 − 1.12i)15-s + (−0.444 − 0.769i)16-s + (0.462 − 0.123i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 + 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.810 + 0.585i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.810 + 0.585i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.839230 - 0.271637i\)
\(L(\frac12)\) \(\approx\) \(0.839230 - 0.271637i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.45 + 1.69i)T \)
13 \( 1 + (1.35 - 3.34i)T \)
good2 \( 1 + (0.237 - 0.137i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.611 + 2.28i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (0.193 - 0.334i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.12 - 4.21i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.90 + 0.510i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (4.83 - 1.29i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.322 - 0.0863i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-7.07 + 4.08i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.54 - 2.54i)T - 31iT^{2} \)
37 \( 1 + (2.41 + 4.17i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.49 - 1.20i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.76 + 6.58i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + 9.83T + 47T^{2} \)
53 \( 1 + (7.17 + 7.17i)T + 53iT^{2} \)
59 \( 1 + (0.628 + 2.34i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (5.32 - 9.22i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.52 + 3.18i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.12 + 4.20i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 - 6.08iT - 73T^{2} \)
79 \( 1 - 3.34iT - 79T^{2} \)
83 \( 1 - 5.18T + 83T^{2} \)
89 \( 1 + (-4.82 - 1.29i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-12.7 - 7.37i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.39283924735611274553777720797, −13.46339935611751821274948283290, −12.54549455108943206584285977656, −12.21268463114154310224424941939, −9.856896343543824856809586617313, −8.750637658077539008738670875183, −7.73757990537169006112394637857, −6.63522658477628275099557837842, −4.62618839766855619027305836632, −2.13492692750337816935576321883, 3.15299074418077516487611253347, 4.95349862258932626229744721669, 6.16688214634064142278114116429, 8.419664621509037730896008883344, 9.559414067395748037187279116498, 10.43775148195440679880050966763, 10.89840996118797164530860440018, 13.17426820957693504419246506626, 14.27392724874111161675521391470, 14.87694819067677098245980512601

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