Properties

Label 2-65-65.37-c1-0-3
Degree $2$
Conductor $65$
Sign $0.887 + 0.460i$
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  + (1.37 − 0.792i)2-s + (0.190 + 0.0510i)3-s + (0.255 − 0.442i)4-s + (−2.23 + 0.0672i)5-s + (0.302 − 0.0809i)6-s + (0.274 − 0.474i)7-s + 2.35i·8-s + (−2.56 − 1.48i)9-s + (−3.01 + 1.86i)10-s + (0.147 + 0.0396i)11-s + (0.0713 − 0.0713i)12-s + (3.21 − 1.63i)13-s − 0.868i·14-s + (−0.429 − 0.101i)15-s + (2.38 + 4.12i)16-s + (0.813 + 3.03i)17-s + ⋯
L(s)  = 1  + (0.970 − 0.560i)2-s + (0.110 + 0.0294i)3-s + (0.127 − 0.221i)4-s + (−0.999 + 0.0300i)5-s + (0.123 − 0.0330i)6-s + (0.103 − 0.179i)7-s + 0.834i·8-s + (−0.854 − 0.493i)9-s + (−0.953 + 0.589i)10-s + (0.0446 + 0.0119i)11-s + (0.0205 − 0.0205i)12-s + (0.891 − 0.452i)13-s − 0.232i·14-s + (−0.110 − 0.0261i)15-s + (0.595 + 1.03i)16-s + (0.197 + 0.736i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.887 + 0.460i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.887 + 0.460i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17838 - 0.287708i\)
\(L(\frac12)\) \(\approx\) \(1.17838 - 0.287708i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.23 - 0.0672i)T \)
13 \( 1 + (-3.21 + 1.63i)T \)
good2 \( 1 + (-1.37 + 0.792i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.190 - 0.0510i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.274 + 0.474i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.147 - 0.0396i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.813 - 3.03i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.18 + 4.40i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.916 - 3.41i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.02 + 1.17i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.60 + 6.60i)T + 31iT^{2} \)
37 \( 1 + (-3.40 - 5.89i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.926 + 3.45i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.86 + 1.84i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 9.13T + 47T^{2} \)
53 \( 1 + (3.70 - 3.70i)T - 53iT^{2} \)
59 \( 1 + (3.67 - 0.985i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.92 - 6.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.23 + 2.44i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (15.1 - 4.04i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + 3.91iT - 73T^{2} \)
79 \( 1 - 11.1iT - 79T^{2} \)
83 \( 1 - 13.4T + 83T^{2} \)
89 \( 1 + (-2.35 + 8.78i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-6.55 - 3.78i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.72200427236009071208582969873, −13.58485565413118906331508316747, −12.61380171473156450018371789300, −11.54936333519394988461906092421, −10.91874375732322795523875925969, −8.859015407758667393937851748816, −7.82573989330391354950218101819, −5.89817500434243568027902757734, −4.24779354124363893153790700959, −3.16240660010681108494101455996, 3.54693171641856956173840258619, 4.93249721495282095080377727041, 6.27909311816704690903038170840, 7.71047620558244841052222096669, 8.930451355069026022130602246212, 10.74463841437163010078892518825, 11.89068321150659794666914241304, 12.94867237038030213146195614604, 14.24676227405272431716752403664, 14.65334204770452539710577200798

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