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Properties

Label 2-65-65.8-c1-0-4
Degree $2$
Conductor $65$
Sign $-0.277 + 0.960i$
Analytic cond. $0.519027$
Root an. cond. $0.720435$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.134·2-s + (−2.15 − 2.15i)3-s − 1.98·4-s + (1.82 − 1.29i)5-s + (0.290 + 0.290i)6-s − 1.90i·7-s + 0.536·8-s + 6.29i·9-s + (−0.245 + 0.173i)10-s + (−0.290 + 0.290i)11-s + (4.27 + 4.27i)12-s + (3.60 + 0.173i)13-s + 0.257i·14-s + (−6.71 − 1.15i)15-s + 3.89·16-s + (−2.53 − 2.53i)17-s + ⋯
L(s)  = 1  − 0.0951·2-s + (−1.24 − 1.24i)3-s − 0.990·4-s + (0.816 − 0.576i)5-s + (0.118 + 0.118i)6-s − 0.721i·7-s + 0.189·8-s + 2.09i·9-s + (−0.0777 + 0.0549i)10-s + (−0.0875 + 0.0875i)11-s + (1.23 + 1.23i)12-s + (0.998 + 0.0481i)13-s + 0.0687i·14-s + (−1.73 − 0.298i)15-s + 0.972·16-s + (−0.615 − 0.615i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.277 + 0.960i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ -0.277 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.332904 - 0.442454i\)
\(L(\frac12)\) \(\approx\) \(0.332904 - 0.442454i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.82 + 1.29i)T \)
13 \( 1 + (-3.60 - 0.173i)T \)
good2 \( 1 + 0.134T + 2T^{2} \)
3 \( 1 + (2.15 + 2.15i)T + 3iT^{2} \)
7 \( 1 + 1.90iT - 7T^{2} \)
11 \( 1 + (0.290 - 0.290i)T - 11iT^{2} \)
17 \( 1 + (2.53 + 2.53i)T + 17iT^{2} \)
19 \( 1 + (-3.15 + 3.15i)T - 19iT^{2} \)
23 \( 1 + (2.27 - 2.27i)T - 23iT^{2} \)
29 \( 1 - 2.40iT - 29T^{2} \)
31 \( 1 + (-2.02 - 2.02i)T + 31iT^{2} \)
37 \( 1 - 5.32iT - 37T^{2} \)
41 \( 1 + (1.51 + 1.51i)T + 41iT^{2} \)
43 \( 1 + (-0.888 + 0.888i)T - 43iT^{2} \)
47 \( 1 - 6.94iT - 47T^{2} \)
53 \( 1 + (1.09 + 1.09i)T + 53iT^{2} \)
59 \( 1 + (-8.31 - 8.31i)T + 59iT^{2} \)
61 \( 1 - 7.17T + 61T^{2} \)
67 \( 1 + 0.939T + 67T^{2} \)
71 \( 1 + (7.37 + 7.37i)T + 71iT^{2} \)
73 \( 1 + 6.63T + 73T^{2} \)
79 \( 1 + 4.39iT - 79T^{2} \)
83 \( 1 - 13.4iT - 83T^{2} \)
89 \( 1 + (10.0 + 10.0i)T + 89iT^{2} \)
97 \( 1 + 4.39T + 97T^{2} \)
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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

−13.73198301886497196750279466565, −13.56517512055824109478521904972, −12.59110053615028309535190585306, −11.35196522802303792416459528823, −10.09985069496473112829791244146, −8.677998386283046688142644055334, −7.21935838656273575774134866803, −5.90696980874424973850839559141, −4.79025585203389526205423182940, −1.08644087346231578302571292467, 3.90845422570094098027227911901, 5.41496974358630568270895729773, 6.13441084724006496807754619305, 8.694235655361333921412445203602, 9.777926995103257129221057698187, 10.50142200532292456655426604940, 11.62700329579126878421940178074, 13.00407925198356294859840901796, 14.28359693793779263197893418039, 15.33900117668743132902221213371

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