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Properties

Label 2-65-13.7-c2-0-4
Degree $2$
Conductor $65$
Sign $0.775 - 0.631i$
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  + (−2.29 + 0.614i)2-s + (0.609 − 1.05i)3-s + (1.41 − 0.815i)4-s + (1.58 + 1.58i)5-s + (−0.748 + 2.79i)6-s + (4.98 + 1.33i)7-s + (3.97 − 3.97i)8-s + (3.75 + 6.50i)9-s + (−4.59 − 2.65i)10-s + (4.82 + 18.0i)11-s − 1.98i·12-s + (7.88 − 10.3i)13-s − 12.2·14-s + (2.63 − 0.705i)15-s + (−9.93 + 17.2i)16-s + (−8.39 + 4.84i)17-s + ⋯
L(s)  = 1  + (−1.14 + 0.307i)2-s + (0.203 − 0.351i)3-s + (0.352 − 0.203i)4-s + (0.316 + 0.316i)5-s + (−0.124 + 0.465i)6-s + (0.711 + 0.190i)7-s + (0.497 − 0.497i)8-s + (0.417 + 0.722i)9-s + (−0.459 − 0.265i)10-s + (0.438 + 1.63i)11-s − 0.165i·12-s + (0.606 − 0.794i)13-s − 0.874·14-s + (0.175 − 0.0470i)15-s + (−0.620 + 1.07i)16-s + (−0.494 + 0.285i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.775 - 0.631i$
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: \((2,\ 65,\ (\ :1),\ 0.775 - 0.631i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.768267 + 0.273060i\)
\(L(\frac12)\) \(\approx\) \(0.768267 + 0.273060i\)
\(L(2)\) 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 \( 1 + (-7.88 + 10.3i)T \)
good2 \( 1 + (2.29 - 0.614i)T + (3.46 - 2i)T^{2} \)
3 \( 1 + (-0.609 + 1.05i)T + (-4.5 - 7.79i)T^{2} \)
7 \( 1 + (-4.98 - 1.33i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (-4.82 - 18.0i)T + (-104. + 60.5i)T^{2} \)
17 \( 1 + (8.39 - 4.84i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-8.46 + 31.5i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (21.3 + 12.3i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (15.4 - 26.6i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (1.39 + 1.39i)T + 961iT^{2} \)
37 \( 1 + (-3.49 - 13.0i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-0.453 + 0.121i)T + (1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-5.09 + 2.94i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-40.0 + 40.0i)T - 2.20e3iT^{2} \)
53 \( 1 + 58.1T + 2.80e3T^{2} \)
59 \( 1 + (-56.9 - 15.2i)T + (3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (50.8 + 88.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (64.8 - 17.3i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-10.4 + 39.0i)T + (-4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (21.2 - 21.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 116.T + 6.24e3T^{2} \)
83 \( 1 + (71.3 + 71.3i)T + 6.88e3iT^{2} \)
89 \( 1 + (27.5 + 102. i)T + (-6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (24.6 - 92.0i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

−14.98989501514585023028763167324, −13.63070322786102372598378347738, −12.67492391301998956849622446146, −10.99774693819824137813902694698, −10.02193863806143740211367286653, −8.851984730893953289453444859489, −7.72349300154639709837856186582, −6.84877261985611152038506200508, −4.69050544625744721809387307437, −1.84917281631250461180353101983, 1.34237683398953513937496316476, 4.01933464493958636448293153229, 5.96743955818179525327975542743, 7.925031351907653760069580395635, 8.898203672656210496176867854517, 9.704435743488861924638987745539, 10.93981865089715691190059640001, 11.84038693319480433528177199555, 13.72499936370310378273462787420, 14.32225583088085716674959048475

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