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Properties

Label 2-65-13.7-c2-0-5
Degree $2$
Conductor $65$
Sign $0.988 - 0.152i$
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  + (3.63 − 0.974i)2-s + (−2.58 + 4.47i)3-s + (8.81 − 5.09i)4-s + (−1.58 − 1.58i)5-s + (−5.03 + 18.8i)6-s + (−0.893 − 0.239i)7-s + (16.4 − 16.4i)8-s + (−8.86 − 15.3i)9-s + (−7.29 − 4.21i)10-s + (−0.879 − 3.28i)11-s + 52.6i·12-s + (−12.0 + 4.93i)13-s − 3.48·14-s + (11.1 − 2.99i)15-s + (23.4 − 40.6i)16-s + (−9.86 + 5.69i)17-s + ⋯
L(s)  = 1  + (1.81 − 0.487i)2-s + (−0.861 + 1.49i)3-s + (2.20 − 1.27i)4-s + (−0.316 − 0.316i)5-s + (−0.839 + 3.13i)6-s + (−0.127 − 0.0341i)7-s + (2.05 − 2.05i)8-s + (−0.985 − 1.70i)9-s + (−0.729 − 0.421i)10-s + (−0.0799 − 0.298i)11-s + 4.38i·12-s + (−0.925 + 0.379i)13-s − 0.248·14-s + (0.744 − 0.199i)15-s + (1.46 − 2.54i)16-s + (−0.580 + 0.334i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.988 - 0.152i$
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: \((2,\ 65,\ (\ :1),\ 0.988 - 0.152i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.28456 + 0.175185i\)
\(L(\frac12)\) \(\approx\) \(2.28456 + 0.175185i\)
\(L(2)\) 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 \( 1 + (12.0 - 4.93i)T \)
good2 \( 1 + (-3.63 + 0.974i)T + (3.46 - 2i)T^{2} \)
3 \( 1 + (2.58 - 4.47i)T + (-4.5 - 7.79i)T^{2} \)
7 \( 1 + (0.893 + 0.239i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (0.879 + 3.28i)T + (-104. + 60.5i)T^{2} \)
17 \( 1 + (9.86 - 5.69i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-3.16 + 11.7i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (-33.9 - 19.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (5.86 - 10.1i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-18.4 - 18.4i)T + 961iT^{2} \)
37 \( 1 + (-3.90 - 14.5i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-9.84 + 2.63i)T + (1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (29.0 - 16.7i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-40.8 + 40.8i)T - 2.20e3iT^{2} \)
53 \( 1 + 21.3T + 2.80e3T^{2} \)
59 \( 1 + (10.2 + 2.73i)T + (3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (47.5 + 82.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (25.7 - 6.90i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (10.2 - 38.0i)T + (-4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-89.2 + 89.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 8.13T + 6.24e3T^{2} \)
83 \( 1 + (-46.6 - 46.6i)T + 6.88e3iT^{2} \)
89 \( 1 + (-10.0 - 37.6i)T + (-6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (36.4 - 135. i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

−14.87885625154637538685722726704, −13.51646144765541199020893041770, −12.31084907420527803730062683893, −11.40281277288576320837176159693, −10.72698888773893489828480562989, −9.453693080380853978836294010449, −6.69089586799597816764722960441, −5.24885288112514425997588935739, −4.62327829595302137441813596013, −3.32680634693287302673160633430, 2.59996335585522811707692378872, 4.79418196970127224752880160659, 6.03621039169248932197333043785, 6.99496830742364977924759069101, 7.73304416425425866287523946203, 10.92070011850028141895715234400, 11.89022419023378022082960702016, 12.58929198746154700063092769035, 13.30320254755214621669299783524, 14.38722242627268555702086251099

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