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Properties

Label 2-65-65.18-c1-0-4
Degree $2$
Conductor $65$
Sign $-0.623 + 0.781i$
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  − 2.03i·2-s + (−1.32 − 1.32i)3-s − 2.12·4-s + (1.70 + 1.45i)5-s + (−2.70 + 2.70i)6-s − 1.61·7-s + 0.248i·8-s + 0.537i·9-s + (2.94 − 3.45i)10-s + (2.70 + 2.70i)11-s + (2.82 + 2.82i)12-s + (3.45 − 1.04i)13-s + 3.28i·14-s + (−0.329 − 4.19i)15-s − 3.74·16-s + (2.24 + 2.24i)17-s + ⋯
L(s)  = 1  − 1.43i·2-s + (−0.767 − 0.767i)3-s − 1.06·4-s + (0.760 + 0.649i)5-s + (−1.10 + 1.10i)6-s − 0.611·7-s + 0.0877i·8-s + 0.179i·9-s + (0.932 − 1.09i)10-s + (0.814 + 0.814i)11-s + (0.814 + 0.814i)12-s + (0.957 − 0.288i)13-s + 0.878i·14-s + (−0.0852 − 1.08i)15-s − 0.935·16-s + (0.545 + 0.545i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.623 + 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.623 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.623 + 0.781i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ -0.623 + 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.340772 - 0.708103i\)
\(L(\frac12)\) \(\approx\) \(0.340772 - 0.708103i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.70 - 1.45i)T \)
13 \( 1 + (-3.45 + 1.04i)T \)
good2 \( 1 + 2.03iT - 2T^{2} \)
3 \( 1 + (1.32 + 1.32i)T + 3iT^{2} \)
7 \( 1 + 1.61T + 7T^{2} \)
11 \( 1 + (-2.70 - 2.70i)T + 11iT^{2} \)
17 \( 1 + (-2.24 - 2.24i)T + 17iT^{2} \)
19 \( 1 + (2.32 + 2.32i)T + 19iT^{2} \)
23 \( 1 + (4.82 - 4.82i)T - 23iT^{2} \)
29 \( 1 - 4.27iT - 29T^{2} \)
31 \( 1 + (-3.36 + 3.36i)T - 31iT^{2} \)
37 \( 1 + 7.78T + 37T^{2} \)
41 \( 1 + (-2.87 + 2.87i)T - 41iT^{2} \)
43 \( 1 + (-3.97 + 3.97i)T - 43iT^{2} \)
47 \( 1 - 5.36T + 47T^{2} \)
53 \( 1 + (4.61 + 4.61i)T + 53iT^{2} \)
59 \( 1 + (-4.47 + 4.47i)T - 59iT^{2} \)
61 \( 1 + 12.1T + 61T^{2} \)
67 \( 1 + 5.84iT - 67T^{2} \)
71 \( 1 + (1.37 - 1.37i)T - 71iT^{2} \)
73 \( 1 - 4.02iT - 73T^{2} \)
79 \( 1 - 8.63iT - 79T^{2} \)
83 \( 1 + 7.48T + 83T^{2} \)
89 \( 1 + (-8.59 + 8.59i)T - 89iT^{2} \)
97 \( 1 + 8.63iT - 97T^{2} \)
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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

−13.98617419971377400260289321904, −12.95769456045494488029736453186, −12.25222014045571956385294201847, −11.23455374282910564260224919705, −10.25513166358890393908697194357, −9.273295390443282838754216440067, −6.95064798054050482535908870810, −6.00339715772345686883820357953, −3.58694210607745382157200566185, −1.68593253710102619942642065110, 4.41656751633675708093808512305, 5.84079827862216822915270236847, 6.31482594562809561032123816117, 8.291742365793465681924367972210, 9.332671756553578007380201814308, 10.57746014219419102711316624116, 11.99371163031034591267071838237, 13.58171575303077116155031150171, 14.28267113154988779127205402758, 15.84918549354769909990206518714

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