Properties

Label 2-1840-460.439-c0-0-1
Degree $2$
Conductor $1840$
Sign $0.320 + 0.947i$
Analytic cond. $0.918279$
Root an. cond. $0.958269$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.153 − 1.07i)3-s + (0.959 + 0.281i)5-s + (0.989 − 1.14i)7-s + (−0.162 + 0.0476i)9-s + (0.153 − 1.07i)15-s + (−1.37 − 0.883i)21-s + (−0.540 + 0.841i)23-s + (0.841 + 0.540i)25-s + (−0.373 − 0.817i)27-s + (−0.797 + 1.74i)29-s + (1.27 − 0.817i)35-s + (−1.25 − 0.368i)41-s + (0.258 + 1.80i)43-s − 0.169·45-s + 0.563·47-s + ⋯
L(s)  = 1  + (−0.153 − 1.07i)3-s + (0.959 + 0.281i)5-s + (0.989 − 1.14i)7-s + (−0.162 + 0.0476i)9-s + (0.153 − 1.07i)15-s + (−1.37 − 0.883i)21-s + (−0.540 + 0.841i)23-s + (0.841 + 0.540i)25-s + (−0.373 − 0.817i)27-s + (−0.797 + 1.74i)29-s + (1.27 − 0.817i)35-s + (−1.25 − 0.368i)41-s + (0.258 + 1.80i)43-s − 0.169·45-s + 0.563·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.320 + 0.947i$
Analytic conductor: \(0.918279\)
Root analytic conductor: \(0.958269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :0),\ 0.320 + 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.445733085\)
\(L(\frac12)\) \(\approx\) \(1.445733085\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.959 - 0.281i)T \)
23 \( 1 + (0.540 - 0.841i)T \)
good3 \( 1 + (0.153 + 1.07i)T + (-0.959 + 0.281i)T^{2} \)
7 \( 1 + (-0.989 + 1.14i)T + (-0.142 - 0.989i)T^{2} \)
11 \( 1 + (-0.415 + 0.909i)T^{2} \)
13 \( 1 + (0.142 - 0.989i)T^{2} \)
17 \( 1 + (0.654 + 0.755i)T^{2} \)
19 \( 1 + (0.654 - 0.755i)T^{2} \)
29 \( 1 + (0.797 - 1.74i)T + (-0.654 - 0.755i)T^{2} \)
31 \( 1 + (0.959 + 0.281i)T^{2} \)
37 \( 1 + (-0.841 + 0.540i)T^{2} \)
41 \( 1 + (1.25 + 0.368i)T + (0.841 + 0.540i)T^{2} \)
43 \( 1 + (-0.258 - 1.80i)T + (-0.959 + 0.281i)T^{2} \)
47 \( 1 - 0.563T + T^{2} \)
53 \( 1 + (0.142 + 0.989i)T^{2} \)
59 \( 1 + (0.142 - 0.989i)T^{2} \)
61 \( 1 + (-0.118 + 0.822i)T + (-0.959 - 0.281i)T^{2} \)
67 \( 1 + (1.66 + 1.07i)T + (0.415 + 0.909i)T^{2} \)
71 \( 1 + (-0.415 - 0.909i)T^{2} \)
73 \( 1 + (0.654 - 0.755i)T^{2} \)
79 \( 1 + (0.142 - 0.989i)T^{2} \)
83 \( 1 + (1.89 - 0.557i)T + (0.841 - 0.540i)T^{2} \)
89 \( 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2} \)
97 \( 1 + (-0.841 - 0.540i)T^{2} \)
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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

−9.324805372262467202573093883310, −8.302260250496499263867260341569, −7.45865485235614553345340265292, −7.08417946948941614662514138237, −6.22483660108119053215625012217, −5.37171931768850643364114358788, −4.43806473789878155493725241975, −3.25084152502315664631982213082, −1.81989239485359275316367295905, −1.34807238197414105560840664360, 1.76968567652109816844179979222, 2.58138768024164921070491918371, 4.04415993290332208602055297269, 4.76660639554355887987378909510, 5.53556986014992098027614595452, 5.98880476706943122174815423515, 7.25625252915749256246376191318, 8.423056818248937688041328769795, 8.821495444742304678787638383527, 9.690052450764438965864545658169

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