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Properties

Label 2-65-13.2-c2-0-1
Degree $2$
Conductor $65$
Sign $0.275 + 0.961i$
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  + (−3.77 − 1.01i)2-s + (−0.338 − 0.585i)3-s + (9.76 + 5.63i)4-s + (−1.58 + 1.58i)5-s + (0.684 + 2.55i)6-s + (4.53 − 1.21i)7-s + (−20.1 − 20.1i)8-s + (4.27 − 7.39i)9-s + (7.56 − 4.37i)10-s + (1.84 − 6.88i)11-s − 7.63i·12-s + (−0.564 − 12.9i)13-s − 18.3·14-s + (1.46 + 0.391i)15-s + (33.0 + 57.2i)16-s + (16.9 + 9.76i)17-s + ⋯
L(s)  = 1  + (−1.88 − 0.505i)2-s + (−0.112 − 0.195i)3-s + (2.44 + 1.40i)4-s + (−0.316 + 0.316i)5-s + (0.114 + 0.425i)6-s + (0.647 − 0.173i)7-s + (−2.51 − 2.51i)8-s + (0.474 − 0.821i)9-s + (0.756 − 0.437i)10-s + (0.167 − 0.625i)11-s − 0.635i·12-s + (−0.0434 − 0.999i)13-s − 1.31·14-s + (0.0974 + 0.0261i)15-s + (2.06 + 3.57i)16-s + (0.994 + 0.574i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.275 + 0.961i$
Analytic conductor: \(1.77112\)
Root analytic conductor: \(1.33083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1),\ 0.275 + 0.961i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.425133 - 0.320290i\)
\(L(\frac12)\) \(\approx\) \(0.425133 - 0.320290i\)
\(L(2)\) 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 \( 1 + (0.564 + 12.9i)T \)
good2 \( 1 + (3.77 + 1.01i)T + (3.46 + 2i)T^{2} \)
3 \( 1 + (0.338 + 0.585i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + (-4.53 + 1.21i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (-1.84 + 6.88i)T + (-104. - 60.5i)T^{2} \)
17 \( 1 + (-16.9 - 9.76i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (2.43 + 9.10i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (-21.0 + 12.1i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (0.898 + 1.55i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-14.7 + 14.7i)T - 961iT^{2} \)
37 \( 1 + (15.2 - 57.0i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (31.5 + 8.44i)T + (1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (3.08 + 1.78i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-41.9 - 41.9i)T + 2.20e3iT^{2} \)
53 \( 1 + 79.2T + 2.80e3T^{2} \)
59 \( 1 + (-42.0 + 11.2i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-14.4 + 24.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (5.37 + 1.43i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (-31.4 - 117. i)T + (-4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (22.0 + 22.0i)T + 5.32e3iT^{2} \)
79 \( 1 + 40.0T + 6.24e3T^{2} \)
83 \( 1 + (-102. + 102. i)T - 6.88e3iT^{2} \)
89 \( 1 + (16.6 - 62.1i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-8.92 - 33.3i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−14.90006224539003878886305481727, −12.74581203592729181734445204217, −11.73596335254922243609796150564, −10.82121713891436551493851599911, −9.886252331224680119214003648598, −8.550442547872461245365999572806, −7.67704293395827761223559993868, −6.46829926760799850366738791629, −3.21824240498525428984234754431, −1.00483647001908262548896566930, 1.70693963179103025940297222556, 5.19938261362534426717744506188, 7.02874633493339389448710416536, 7.88152857305207455103564530837, 9.042939874167029390743058936636, 10.03463566517236168525907257143, 11.12766483017943334573244058825, 12.07593790047878650744068212108, 14.29126576720832881547010941951, 15.34472303471795655975531349495

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