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Properties

Label 2-65-65.12-c2-0-9
Degree $2$
Conductor $65$
Sign $0.286 + 0.958i$
Analytic cond. $1.77112$
Root an. cond. $1.33083$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.78 − 1.78i)2-s + (0.433 − 0.433i)3-s − 2.33i·4-s + (−3.33 − 3.72i)5-s − 1.54i·6-s + (5.63 − 5.63i)7-s + (2.95 + 2.95i)8-s + 8.62i·9-s + (−12.5 − 0.693i)10-s + 7.95i·11-s + (−1.01 − 1.01i)12-s + (−10.8 + 7.09i)13-s − 20.0i·14-s + (−3.06 − 0.169i)15-s + 19.8·16-s + (4.22 + 4.22i)17-s + ⋯
L(s)  = 1  + (0.890 − 0.890i)2-s + (0.144 − 0.144i)3-s − 0.584i·4-s + (−0.667 − 0.744i)5-s − 0.257i·6-s + (0.805 − 0.805i)7-s + (0.369 + 0.369i)8-s + 0.958i·9-s + (−1.25 − 0.0693i)10-s + 0.723i·11-s + (−0.0845 − 0.0845i)12-s + (−0.837 + 0.545i)13-s − 1.43i·14-s + (−0.204 − 0.0112i)15-s + 1.24·16-s + (0.248 + 0.248i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.286 + 0.958i$
Analytic conductor: \(1.77112\)
Root analytic conductor: \(1.33083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1),\ 0.286 + 0.958i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.45970 - 1.08740i\)
\(L(\frac12)\) \(\approx\) \(1.45970 - 1.08740i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (3.33 + 3.72i)T \)
13 \( 1 + (10.8 - 7.09i)T \)
good2 \( 1 + (-1.78 + 1.78i)T - 4iT^{2} \)
3 \( 1 + (-0.433 + 0.433i)T - 9iT^{2} \)
7 \( 1 + (-5.63 + 5.63i)T - 49iT^{2} \)
11 \( 1 - 7.95iT - 121T^{2} \)
17 \( 1 + (-4.22 - 4.22i)T + 289iT^{2} \)
19 \( 1 + 28.4T + 361T^{2} \)
23 \( 1 + (-16.8 + 16.8i)T - 529iT^{2} \)
29 \( 1 + 50.7iT - 841T^{2} \)
31 \( 1 - 5.26iT - 961T^{2} \)
37 \( 1 + (16.4 - 16.4i)T - 1.36e3iT^{2} \)
41 \( 1 + 61.6iT - 1.68e3T^{2} \)
43 \( 1 + (18.6 - 18.6i)T - 1.84e3iT^{2} \)
47 \( 1 + (21.8 - 21.8i)T - 2.20e3iT^{2} \)
53 \( 1 + (-38.2 + 38.2i)T - 2.80e3iT^{2} \)
59 \( 1 + 12.8T + 3.48e3T^{2} \)
61 \( 1 + 45.4T + 3.72e3T^{2} \)
67 \( 1 + (22.7 - 22.7i)T - 4.48e3iT^{2} \)
71 \( 1 - 0.496iT - 5.04e3T^{2} \)
73 \( 1 + (18.8 + 18.8i)T + 5.32e3iT^{2} \)
79 \( 1 - 73.0iT - 6.24e3T^{2} \)
83 \( 1 + (-70.8 - 70.8i)T + 6.88e3iT^{2} \)
89 \( 1 - 2.26T + 7.92e3T^{2} \)
97 \( 1 + (-85.2 + 85.2i)T - 9.40e3iT^{2} \)
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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

−14.13862512337717767849175037309, −13.12663240298800469065952681423, −12.31163208791397677110694594145, −11.29442888514575441346288505685, −10.32582807983120778493578034336, −8.367010841430633666397198927814, −7.42467856977811765631138165495, −4.83231455047685132388317265452, −4.27392218021366510191155207234, −2.02545091069421763656510853621, 3.34538986998285887945821344808, 4.93319843104531954536393508376, 6.25333347596910900676420064168, 7.44742950780327269416853229164, 8.738621307612723058153929892924, 10.48345906839657931390489890297, 11.74141334619314621179118240153, 12.79924823065031263603811529096, 14.31164926146225787188175275156, 14.96547847472254401697896201527

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