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Properties

Label 2-65-65.63-c1-0-4
Degree $2$
Conductor $65$
Sign $-0.681 + 0.731i$
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  + (−1.04 − 1.80i)2-s + (0.713 − 2.66i)3-s + (−1.17 + 2.04i)4-s + (−0.194 + 2.22i)5-s + (−5.55 + 1.48i)6-s + (2.52 + 1.45i)7-s + 0.750·8-s + (−3.97 − 2.29i)9-s + (4.23 − 1.97i)10-s + (−0.0254 − 0.00681i)11-s + (4.59 + 4.59i)12-s + (0.530 − 3.56i)13-s − 6.07i·14-s + (5.79 + 2.10i)15-s + (1.57 + 2.72i)16-s + (−2.76 + 0.741i)17-s + ⋯
L(s)  = 1  + (−0.738 − 1.27i)2-s + (0.411 − 1.53i)3-s + (−0.589 + 1.02i)4-s + (−0.0869 + 0.996i)5-s + (−2.26 + 0.607i)6-s + (0.952 + 0.550i)7-s + 0.265·8-s + (−1.32 − 0.765i)9-s + (1.33 − 0.624i)10-s + (−0.00767 − 0.00205i)11-s + (1.32 + 1.32i)12-s + (0.147 − 0.989i)13-s − 1.62i·14-s + (1.49 + 0.543i)15-s + (0.393 + 0.682i)16-s + (−0.671 + 0.179i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.681 + 0.731i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ -0.681 + 0.731i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.288408 - 0.662536i\)
\(L(\frac12)\) \(\approx\) \(0.288408 - 0.662536i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.194 - 2.22i)T \)
13 \( 1 + (-0.530 + 3.56i)T \)
good2 \( 1 + (1.04 + 1.80i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.713 + 2.66i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-2.52 - 1.45i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.0254 + 0.00681i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.76 - 0.741i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.23 - 4.62i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.358 + 0.0961i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-3.62 + 2.09i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.835 + 0.835i)T + 31iT^{2} \)
37 \( 1 + (5.58 - 3.22i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.02 - 7.57i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.79 - 6.69i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + 0.833iT - 47T^{2} \)
53 \( 1 + (-0.902 - 0.902i)T + 53iT^{2} \)
59 \( 1 + (-1.44 + 0.387i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-5.35 + 9.26i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.15 + 10.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.57 - 0.957i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + 15.0T + 73T^{2} \)
79 \( 1 - 4.25iT - 79T^{2} \)
83 \( 1 - 1.31iT - 83T^{2} \)
89 \( 1 + (-0.867 + 3.23i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.202 + 0.351i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.26222777389718041028905432252, −13.06062732692285471687530247716, −12.03891898047852665865792609293, −11.30109188108949723963241917198, −10.15140274205348406599587074621, −8.478314970286155156749042545344, −7.79024183846667897223540886070, −6.20293851171000530096100560607, −2.99936682645872709250941308262, −1.77338902645494238635869283829, 4.29333790168640961448416100117, 5.24319573814241861289616450755, 7.24108847402340606747155731919, 8.778269383076929695978787742773, 8.943262434112117178935477684653, 10.33408266268316920453396499862, 11.66335470124702087702928189285, 13.76043526295930955470271113603, 14.63057342171884284693885707143, 15.73505698283781992707400284604

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