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Properties

Label 2-65-65.47-c1-0-0
Degree $2$
Conductor $65$
Sign $0.392 - 0.919i$
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  − 0.134i·2-s + (−2.15 + 2.15i)3-s + 1.98·4-s + (−1.29 + 1.82i)5-s + (0.290 + 0.290i)6-s + 1.90·7-s − 0.536i·8-s − 6.29i·9-s + (0.245 + 0.173i)10-s + (−0.290 + 0.290i)11-s + (−4.27 + 4.27i)12-s + (0.173 − 3.60i)13-s − 0.257i·14-s + (−1.15 − 6.71i)15-s + 3.89·16-s + (2.53 − 2.53i)17-s + ⋯
L(s)  = 1  − 0.0951i·2-s + (−1.24 + 1.24i)3-s + 0.990·4-s + (−0.576 + 0.816i)5-s + (0.118 + 0.118i)6-s + 0.721·7-s − 0.189i·8-s − 2.09i·9-s + (0.0777 + 0.0549i)10-s + (−0.0875 + 0.0875i)11-s + (−1.23 + 1.23i)12-s + (0.0481 − 0.998i)13-s − 0.0687i·14-s + (−0.298 − 1.73i)15-s + 0.972·16-s + (0.615 − 0.615i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.392 - 0.919i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.392 - 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.625085 + 0.412726i\)
\(L(\frac12)\) \(\approx\) \(0.625085 + 0.412726i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.29 - 1.82i)T \)
13 \( 1 + (-0.173 + 3.60i)T \)
good2 \( 1 + 0.134iT - 2T^{2} \)
3 \( 1 + (2.15 - 2.15i)T - 3iT^{2} \)
7 \( 1 - 1.90T + 7T^{2} \)
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 \( 1 + 5.32T + 37T^{2} \)
41 \( 1 + (1.51 + 1.51i)T + 41iT^{2} \)
43 \( 1 + (0.888 + 0.888i)T + 43iT^{2} \)
47 \( 1 + 6.94T + 47T^{2} \)
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 \( 1 + 0.939iT - 67T^{2} \)
71 \( 1 + (7.37 + 7.37i)T + 71iT^{2} \)
73 \( 1 - 6.63iT - 73T^{2} \)
79 \( 1 - 4.39iT - 79T^{2} \)
83 \( 1 - 13.4T + 83T^{2} \)
89 \( 1 + (-10.0 - 10.0i)T + 89iT^{2} \)
97 \( 1 + 4.39iT - 97T^{2} \)
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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

−15.33779505108230426802958076755, −14.60220231323700401069136274213, −12.26305924485290084657774313818, −11.47380142799838086894426330220, −10.72995371838248797446178264687, −10.05688556617846379004864180045, −7.85188043004830227224449801327, −6.42631704887152813800309218563, −5.16352461134965396108962758127, −3.45819930895912012042979199078, 1.59490065267106985890189734947, 4.93592658142669140209693027403, 6.29517315320610496333082381339, 7.32602732443817652526306987509, 8.384673004722584561349944785054, 10.78966592555667943865991507304, 11.58429526682986789137709560303, 12.21881166584567956025662117590, 13.19244407968382834250125356431, 14.78868125359851007184471037943

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