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Properties

Label 2-65-13.12-c1-0-0
Degree $2$
Conductor $65$
Sign $-0.953 - 0.301i$
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  + 2.08i·2-s − 3.08·3-s − 2.35·4-s + i·5-s − 6.43i·6-s + 1.35i·7-s − 0.734i·8-s + 6.52·9-s − 2.08·10-s + 3.73i·11-s + 7.25·12-s + (1.08 − 3.43i)13-s − 2.82·14-s − 3.08i·15-s − 3.17·16-s + 2.70·17-s + ⋯
L(s)  = 1  + 1.47i·2-s − 1.78·3-s − 1.17·4-s + 0.447i·5-s − 2.62i·6-s + 0.510i·7-s − 0.259i·8-s + 2.17·9-s − 0.659·10-s + 1.12i·11-s + 2.09·12-s + (0.301 − 0.953i)13-s − 0.753·14-s − 0.796i·15-s − 0.793·16-s + 0.655·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.301i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 - 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.953 - 0.301i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ -0.953 - 0.301i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0800526 + 0.519146i\)
\(L(\frac12)\) \(\approx\) \(0.0800526 + 0.519146i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - iT \)
13 \( 1 + (-1.08 + 3.43i)T \)
good2 \( 1 - 2.08iT - 2T^{2} \)
3 \( 1 + 3.08T + 3T^{2} \)
7 \( 1 - 1.35iT - 7T^{2} \)
11 \( 1 - 3.73iT - 11T^{2} \)
17 \( 1 - 2.70T + 17T^{2} \)
19 \( 1 - 0.438iT - 19T^{2} \)
23 \( 1 - 5.08T + 23T^{2} \)
29 \( 1 + 1.35T + 29T^{2} \)
31 \( 1 + 6.43iT - 31T^{2} \)
37 \( 1 - 7.35iT - 37T^{2} \)
41 \( 1 - 6.87iT - 41T^{2} \)
43 \( 1 - 0.209T + 43T^{2} \)
47 \( 1 + 1.35iT - 47T^{2} \)
53 \( 1 + 1.46T + 53T^{2} \)
59 \( 1 + 2.26iT - 59T^{2} \)
61 \( 1 - 3.52T + 61T^{2} \)
67 \( 1 + 11.5iT - 67T^{2} \)
71 \( 1 - 0.438iT - 71T^{2} \)
73 \( 1 + 3.69iT - 73T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 + 0.475iT - 83T^{2} \)
89 \( 1 + 11.0iT - 89T^{2} \)
97 \( 1 - 3.29iT - 97T^{2} \)
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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

−15.40052376682719117694690629686, −14.98753707734340996674932495269, −13.18676068250536754945186120706, −12.08104244432529727541911229385, −10.97152075361454116993910809783, −9.734525051350519911706843482592, −7.79155638619228010991487869864, −6.73234369574511293236824621933, −5.79122070531119725550367002383, −4.86321361822595557785108330774, 1.01415196881671313783806455603, 3.96352922720447097511227539398, 5.37098552289758347766473307767, 6.86196605652434717143045465980, 9.133545420011570444519531997438, 10.49779704275395056314411583497, 11.09232241434892556789305546353, 11.92171752506623436926090674785, 12.76465465684772319071237128381, 13.81880083009857469848427496764

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