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Properties

Label 2-65-13.2-c2-0-4
Degree $2$
Conductor $65$
Sign $0.754 - 0.656i$
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  + (2.33 + 0.624i)2-s + (1.05 + 1.82i)3-s + (1.57 + 0.911i)4-s + (−1.58 + 1.58i)5-s + (1.31 + 4.90i)6-s + (2.11 − 0.565i)7-s + (−3.71 − 3.71i)8-s + (2.28 − 3.95i)9-s + (−4.67 + 2.69i)10-s + (1.74 − 6.52i)11-s + 3.83i·12-s + (−11.5 + 6.00i)13-s + 5.27·14-s + (−4.54 − 1.21i)15-s + (−9.98 − 17.2i)16-s + (13.2 + 7.65i)17-s + ⋯
L(s)  = 1  + (1.16 + 0.312i)2-s + (0.350 + 0.607i)3-s + (0.394 + 0.227i)4-s + (−0.316 + 0.316i)5-s + (0.219 + 0.817i)6-s + (0.301 − 0.0808i)7-s + (−0.464 − 0.464i)8-s + (0.254 − 0.439i)9-s + (−0.467 + 0.269i)10-s + (0.159 − 0.593i)11-s + 0.319i·12-s + (−0.886 + 0.461i)13-s + 0.376·14-s + (−0.302 − 0.0811i)15-s + (−0.624 − 1.08i)16-s + (0.779 + 0.450i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.754 - 0.656i$
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: \((2,\ 65,\ (\ :1),\ 0.754 - 0.656i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.92701 + 0.721106i\)
\(L(\frac12)\) \(\approx\) \(1.92701 + 0.721106i\)
\(L(2)\) 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 \( 1 + (11.5 - 6.00i)T \)
good2 \( 1 + (-2.33 - 0.624i)T + (3.46 + 2i)T^{2} \)
3 \( 1 + (-1.05 - 1.82i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + (-2.11 + 0.565i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (-1.74 + 6.52i)T + (-104. - 60.5i)T^{2} \)
17 \( 1 + (-13.2 - 7.65i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (0.594 + 2.21i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (27.2 - 15.7i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-4.77 - 8.27i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (22.6 - 22.6i)T - 961iT^{2} \)
37 \( 1 + (-5.89 + 22.0i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (5.87 + 1.57i)T + (1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (-20.7 - 11.9i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-49.1 - 49.1i)T + 2.20e3iT^{2} \)
53 \( 1 - 93.2T + 2.80e3T^{2} \)
59 \( 1 + (1.51 - 0.405i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-24.6 + 42.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (31.1 + 8.34i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (12.8 + 47.8i)T + (-4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (57.3 + 57.3i)T + 5.32e3iT^{2} \)
79 \( 1 + 142.T + 6.24e3T^{2} \)
83 \( 1 + (-6.65 + 6.65i)T - 6.88e3iT^{2} \)
89 \( 1 + (-43.1 + 160. i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (36.2 + 135. i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−14.50644809516215331905634672670, −14.16781737054612431980774327476, −12.66141840493229324179976112937, −11.74878071714245770046527259006, −10.20016286256929153014046150833, −9.054781174244640745479464651993, −7.35219699557713476034900991530, −5.90890536115259760060706120778, −4.43299558184937674191814034210, −3.38937431348731234134443529895, 2.37997524567137043880867307380, 4.23952870728094493218992617375, 5.44464062889921470524021632728, 7.32200457515584178435465775195, 8.408682686600387504423152646597, 10.10808387030045278455093251481, 11.78998442053251394913317647680, 12.42866458281901449449573614705, 13.33541765241301991702256604095, 14.31479510431116677592613528995

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