↑

Properties

Label 2-65-13.7-c2-0-0
Degree $2$
Conductor $65$
Sign $-0.678 + 0.734i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.54 + 0.683i)2-s + (−2.17 + 3.75i)3-s + (2.57 − 1.48i)4-s + (1.58 + 1.58i)5-s + (2.96 − 11.0i)6-s + (−8.11 − 2.17i)7-s + (1.92 − 1.92i)8-s + (−4.91 − 8.51i)9-s + (−5.11 − 2.95i)10-s + (−3.57 − 13.3i)11-s + 12.8i·12-s + (−6.35 + 11.3i)13-s + 22.1·14-s + (−9.37 + 2.51i)15-s + (−9.53 + 16.5i)16-s + (7.39 − 4.27i)17-s + ⋯
L(s)  = 1  + (−1.27 + 0.341i)2-s + (−0.723 + 1.25i)3-s + (0.642 − 0.371i)4-s + (0.316 + 0.316i)5-s + (0.494 − 1.84i)6-s + (−1.15 − 0.310i)7-s + (0.240 − 0.240i)8-s + (−0.546 − 0.946i)9-s + (−0.511 − 0.295i)10-s + (−0.324 − 1.21i)11-s + 1.07i·12-s + (−0.488 + 0.872i)13-s + 1.58·14-s + (−0.624 + 0.167i)15-s + (−0.595 + 1.03i)16-s + (0.435 − 0.251i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.678 + 0.734i$
Analytic conductor: \(1.77112\)
Root analytic conductor: \(1.33083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1),\ -0.678 + 0.734i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0539024 - 0.123275i\)
\(L(\frac12)\) \(\approx\) \(0.0539024 - 0.123275i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.58 - 1.58i)T \)
13 \( 1 + (6.35 - 11.3i)T \)
good2 \( 1 + (2.54 - 0.683i)T + (3.46 - 2i)T^{2} \)
3 \( 1 + (2.17 - 3.75i)T + (-4.5 - 7.79i)T^{2} \)
7 \( 1 + (8.11 + 2.17i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (3.57 + 13.3i)T + (-104. + 60.5i)T^{2} \)
17 \( 1 + (-7.39 + 4.27i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (3.60 - 13.4i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (18.2 + 10.5i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (27.0 - 46.8i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (15.3 + 15.3i)T + 961iT^{2} \)
37 \( 1 + (1.12 + 4.20i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (4.73 - 1.26i)T + (1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (60.8 - 35.1i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-22.4 + 22.4i)T - 2.20e3iT^{2} \)
53 \( 1 - 50.1T + 2.80e3T^{2} \)
59 \( 1 + (-99.2 - 26.6i)T + (3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-20.1 - 34.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-30.4 + 8.14i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (36.0 - 134. i)T + (-4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (43.7 - 43.7i)T - 5.32e3iT^{2} \)
79 \( 1 + 20.8T + 6.24e3T^{2} \)
83 \( 1 + (50.3 + 50.3i)T + 6.88e3iT^{2} \)
89 \( 1 + (28.5 + 106. i)T + (-6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (-1.04 + 3.90i)T + (-8.14e3 - 4.70e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.15643003064418511682349667133, −14.58739385639340634641931273821, −13.15355103208659919667403284667, −11.40642518008298290413385449498, −10.28078622425604872686410220665, −9.849548632011105611244762132960, −8.739025856530165174842440531767, −7.00873090141972271400598099264, −5.73929495055442476100492830667, −3.78348706361299944858211780784, 0.19082761173302169037556652693, 2.09776809361984841644428893780, 5.51773584903093773160101545462, 6.94743851183491378669669684948, 7.943337237797008181598227451199, 9.498061778212660294632780215307, 10.23495673932002853486310760999, 11.77412950963719811111715506308, 12.65381682095068789605689177355, 13.40982489029392280963383448810

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