Properties

Label 2-65-65.32-c1-0-3
Degree $2$
Conductor $65$
Sign $0.596 + 0.802i$
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  + (1.12 − 1.94i)2-s + (0.514 + 1.91i)3-s + (−1.53 − 2.65i)4-s + (−2.22 − 0.247i)5-s + (4.31 + 1.15i)6-s + (−1.10 + 0.638i)7-s − 2.39·8-s + (−0.820 + 0.473i)9-s + (−2.98 + 4.05i)10-s + (−5.27 + 1.41i)11-s + (4.30 − 4.30i)12-s + (3.50 − 0.840i)13-s + 2.87i·14-s + (−0.666 − 4.39i)15-s + (0.365 − 0.633i)16-s + (3.11 + 0.833i)17-s + ⋯
L(s)  = 1  + (0.795 − 1.37i)2-s + (0.296 + 1.10i)3-s + (−0.766 − 1.32i)4-s + (−0.993 − 0.110i)5-s + (1.76 + 0.472i)6-s + (−0.418 + 0.241i)7-s − 0.848·8-s + (−0.273 + 0.157i)9-s + (−0.943 + 1.28i)10-s + (−1.59 + 0.426i)11-s + (1.24 − 1.24i)12-s + (0.972 − 0.233i)13-s + 0.768i·14-s + (−0.172 − 1.13i)15-s + (0.0913 − 0.158i)16-s + (0.754 + 0.202i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.596 + 0.802i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.596 + 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05209 - 0.528890i\)
\(L(\frac12)\) \(\approx\) \(1.05209 - 0.528890i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.22 + 0.247i)T \)
13 \( 1 + (-3.50 + 0.840i)T \)
good2 \( 1 + (-1.12 + 1.94i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.514 - 1.91i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (1.10 - 0.638i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.27 - 1.41i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-3.11 - 0.833i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.315 + 1.17i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.160 + 0.0428i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (8.41 + 4.85i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.233 + 0.233i)T - 31iT^{2} \)
37 \( 1 + (1.14 + 0.660i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.129 - 0.483i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.72 - 6.43i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 - 3.20iT - 47T^{2} \)
53 \( 1 + (-4.49 + 4.49i)T - 53iT^{2} \)
59 \( 1 + (0.00222 + 0.000595i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.695 + 1.20i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.03 - 5.26i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-11.7 - 3.14i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 - 7.34T + 73T^{2} \)
79 \( 1 + 11.1iT - 79T^{2} \)
83 \( 1 + 2.65iT - 83T^{2} \)
89 \( 1 + (-1.86 - 6.96i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.09 + 3.62i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.79842839305128995833780538080, −13.25858309978173176049927378845, −12.55975698639598009649931313858, −11.27364515380475546288617620923, −10.48844277377549798780080376334, −9.503762359803084872668407745640, −7.902858549223609046143390497729, −5.27622605467119284392717744743, −4.05877097203690452316556916419, −3.06403938626189378884197003228, 3.56328042452326942346974060767, 5.40770898675932553382495091950, 6.83213068706262333605999386208, 7.65246359536913134567674798152, 8.373273502107031791891166909769, 10.74987865572969365470378114257, 12.38131849067684577762253955255, 13.20594260526986949477242247964, 13.87218223261271993366214079877, 15.11609218214936770955610446714

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