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Properties

Label 2-65-13.9-c1-0-2
Degree $2$
Conductor $65$
Sign $0.859 - 0.511i$
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  + (−0.809 + 1.40i)2-s + (1.11 − 1.93i)3-s + (−0.309 − 0.535i)4-s + 5-s + (1.80 + 3.13i)6-s + (0.118 + 0.204i)7-s − 2.23·8-s + (−1 − 1.73i)9-s + (−0.809 + 1.40i)10-s + (−2.11 + 3.66i)11-s − 1.38·12-s + (−1 − 3.46i)13-s − 0.381·14-s + (1.11 − 1.93i)15-s + (2.42 − 4.20i)16-s + (−2.73 − 4.73i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.990i)2-s + (0.645 − 1.11i)3-s + (−0.154 − 0.267i)4-s + 0.447·5-s + (0.738 + 1.27i)6-s + (0.0446 + 0.0772i)7-s − 0.790·8-s + (−0.333 − 0.577i)9-s + (−0.255 + 0.443i)10-s + (−0.638 + 1.10i)11-s − 0.398·12-s + (−0.277 − 0.960i)13-s − 0.102·14-s + (0.288 − 0.500i)15-s + (0.606 − 1.05i)16-s + (−0.663 − 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.809791 + 0.222557i\)
\(L(\frac12)\) \(\approx\) \(0.809791 + 0.222557i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
13 \( 1 + (1 + 3.46i)T \)
good2 \( 1 + (0.809 - 1.40i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.11 + 1.93i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.118 - 0.204i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.11 - 3.66i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.73 + 4.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.118 - 0.204i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.11 - 7.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.736 - 1.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.97 + 5.14i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.881 - 1.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12.9T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (6.35 + 11.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.20 - 10.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.35 - 9.27i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.881 - 1.52i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 8.94T + 83T^{2} \)
89 \( 1 + (-4.5 + 7.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.73 + 4.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−15.15936335559166586063539853301, −13.94672958271611846080473208999, −12.99113616220147973722297728587, −11.99999762178652902333060323061, −10.01009925698353961301550733886, −8.813569820285216209934805029874, −7.58676772867519282729109863211, −7.14178766249817827501905798296, −5.53560787487605899639312025354, −2.50969198149248296814249085003, 2.51624338172632343298722397469, 4.11077043036065636618792526652, 6.09431055068922998204347764681, 8.501118814250077106126849832631, 9.229375035040672833821934710016, 10.36036668894980585211914562316, 10.89200202524759998930627850597, 12.36225849462161726033066145683, 13.83156478238163687988492484012, 14.81102293431810909412345040642

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