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Properties

Label 2-65-13.11-c2-0-0
Degree $2$
Conductor $65$
Sign $-0.964 + 0.265i$
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  + (−1.00 + 3.73i)2-s + (1.63 + 2.83i)3-s + (−9.46 − 5.46i)4-s + (−1.58 − 1.58i)5-s + (−12.2 + 3.27i)6-s + (2.30 + 8.61i)7-s + (18.9 − 18.9i)8-s + (−0.844 + 1.46i)9-s + (7.48 − 4.32i)10-s + (3.75 + 1.00i)11-s − 35.7i·12-s + (−6.85 + 11.0i)13-s − 34.4·14-s + (1.89 − 7.06i)15-s + (29.9 + 51.7i)16-s + (−5.00 − 2.89i)17-s + ⋯
L(s)  = 1  + (−0.500 + 1.86i)2-s + (0.544 + 0.943i)3-s + (−2.36 − 1.36i)4-s + (−0.316 − 0.316i)5-s + (−2.03 + 0.545i)6-s + (0.329 + 1.23i)7-s + (2.36 − 2.36i)8-s + (−0.0938 + 0.162i)9-s + (0.748 − 0.432i)10-s + (0.341 + 0.0913i)11-s − 2.97i·12-s + (−0.527 + 0.849i)13-s − 2.46·14-s + (0.126 − 0.470i)15-s + (1.86 + 3.23i)16-s + (−0.294 − 0.170i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.964 + 0.265i$
Analytic conductor: \(1.77112\)
Root analytic conductor: \(1.33083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1),\ -0.964 + 0.265i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.124492 - 0.921064i\)
\(L(\frac12)\) \(\approx\) \(0.124492 - 0.921064i\)
\(L(2)\) 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 \( 1 + (6.85 - 11.0i)T \)
good2 \( 1 + (1.00 - 3.73i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (-1.63 - 2.83i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + (-2.30 - 8.61i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (-3.75 - 1.00i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (5.00 + 2.89i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-16.7 + 4.48i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (8.48 - 4.89i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (7.28 + 12.6i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-26.8 - 26.8i)T + 961iT^{2} \)
37 \( 1 + (-40.8 - 10.9i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-11.2 + 42.0i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (51.5 + 29.7i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-50.4 + 50.4i)T - 2.20e3iT^{2} \)
53 \( 1 - 28.3T + 2.80e3T^{2} \)
59 \( 1 + (2.48 + 9.25i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-10.4 + 18.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-2.88 + 10.7i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (7.29 - 1.95i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (36.6 - 36.6i)T - 5.32e3iT^{2} \)
79 \( 1 + 45.8T + 6.24e3T^{2} \)
83 \( 1 + (42.4 + 42.4i)T + 6.88e3iT^{2} \)
89 \( 1 + (86.1 + 23.0i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (48.7 - 13.0i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−15.41829105945690289026672278562, −14.69944079529394933617731070004, −13.77214501849185669442647826699, −11.98177615880015127229075359591, −9.869978067716721383064658896726, −9.098296931222761446157198540375, −8.416865793592738614795697690385, −6.97168269778354289760587131114, −5.43290143877832497615526104657, −4.33456708471169992768679895816, 1.09898201355851501760213820261, 2.81540835778374345173757265765, 4.29781306171986655439503511174, 7.49373813394918383744941672859, 8.173545959664315412368139510497, 9.766276065291457584815927996935, 10.71558601450577887611509590605, 11.72198305618664790019904112075, 12.83775091631220283013507220704, 13.57435219334794963821388072657

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