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Properties

Label 2-65-13.3-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  + (0.309 + 0.535i)2-s + (−1.11 − 1.93i)3-s + (0.809 − 1.40i)4-s + 5-s + (0.690 − 1.19i)6-s + (−2.11 + 3.66i)7-s + 2.23·8-s + (−1 + 1.73i)9-s + (0.309 + 0.535i)10-s + (0.118 + 0.204i)11-s − 3.61·12-s + (−1 + 3.46i)13-s − 2.61·14-s + (−1.11 − 1.93i)15-s + (−0.927 − 1.60i)16-s + (1.73 − 3.00i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.378i)2-s + (−0.645 − 1.11i)3-s + (0.404 − 0.700i)4-s + 0.447·5-s + (0.282 − 0.488i)6-s + (−0.800 + 1.38i)7-s + 0.790·8-s + (−0.333 + 0.577i)9-s + (0.0977 + 0.169i)10-s + (0.0355 + 0.0616i)11-s − 1.04·12-s + (−0.277 + 0.960i)13-s − 0.699·14-s + (−0.288 − 0.500i)15-s + (−0.231 − 0.401i)16-s + (0.421 − 0.729i)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: $\chi_{65} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.881842 - 0.242359i\)
\(L(\frac12)\) \(\approx\) \(0.881842 - 0.242359i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 - 0.535i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.11 + 1.93i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (2.11 - 3.66i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.118 - 0.204i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.73 + 3.00i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.11 - 3.66i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.88 + 3.25i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.73 - 6.47i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.97 + 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.11 + 5.40i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.94T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-0.354 + 0.613i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.20 - 12.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.35 - 2.34i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.11 + 5.40i)T + (-35.5 - 61.4i)T^{2} \)
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 \( 1 + (-1.73 + 3.00i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.73300176271884016737820839524, −13.74268997700811759479443586501, −12.41286964325917469070239502382, −11.90051847468747981132561748123, −10.30938188485024706140476848320, −9.015218288918786119772358897591, −7.07440738516544523845869778812, −6.27573316742424405601050745681, −5.40477671216775047552365708067, −2.07564039327805673978410564140, 3.37901253656634988939893715786, 4.59101232351657401245294500722, 6.37010337587138669427635071090, 7.84312595428948964394421230079, 9.884308230922131675060388385853, 10.42568381892046713289286935250, 11.41531999749598155395538623730, 12.85342784185972246918760586522, 13.58824312276214592994386588658, 15.27203221728130883942385367391

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