Properties

Label 2-1840-460.39-c0-0-0
Degree $2$
Conductor $1840$
Sign $-0.130 - 0.991i$
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  + (−1.27 + 0.817i)3-s + (−0.415 − 0.909i)5-s + (−0.540 + 0.158i)7-s + (0.533 − 1.16i)9-s + (1.27 + 0.817i)15-s + (0.557 − 0.643i)21-s + (0.755 + 0.654i)23-s + (−0.654 + 0.755i)25-s + (0.0612 + 0.425i)27-s + (−0.118 + 0.822i)29-s + (0.368 + 0.425i)35-s + (0.797 + 1.74i)41-s + (−1.66 + 1.07i)43-s − 1.28·45-s + 1.81·47-s + ⋯
L(s)  = 1  + (−1.27 + 0.817i)3-s + (−0.415 − 0.909i)5-s + (−0.540 + 0.158i)7-s + (0.533 − 1.16i)9-s + (1.27 + 0.817i)15-s + (0.557 − 0.643i)21-s + (0.755 + 0.654i)23-s + (−0.654 + 0.755i)25-s + (0.0612 + 0.425i)27-s + (−0.118 + 0.822i)29-s + (0.368 + 0.425i)35-s + (0.797 + 1.74i)41-s + (−1.66 + 1.07i)43-s − 1.28·45-s + 1.81·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.130 - 0.991i$
Analytic conductor: \(0.918279\)
Root analytic conductor: \(0.958269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :0),\ -0.130 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4605463227\)
\(L(\frac12)\) \(\approx\) \(0.4605463227\)
\(L(1)\) 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 \( 1 + (0.415 + 0.909i)T \)
23 \( 1 + (-0.755 - 0.654i)T \)
good3 \( 1 + (1.27 - 0.817i)T + (0.415 - 0.909i)T^{2} \)
7 \( 1 + (0.540 - 0.158i)T + (0.841 - 0.540i)T^{2} \)
11 \( 1 + (0.142 - 0.989i)T^{2} \)
13 \( 1 + (-0.841 - 0.540i)T^{2} \)
17 \( 1 + (0.959 + 0.281i)T^{2} \)
19 \( 1 + (0.959 - 0.281i)T^{2} \)
29 \( 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2} \)
31 \( 1 + (-0.415 - 0.909i)T^{2} \)
37 \( 1 + (0.654 + 0.755i)T^{2} \)
41 \( 1 + (-0.797 - 1.74i)T + (-0.654 + 0.755i)T^{2} \)
43 \( 1 + (1.66 - 1.07i)T + (0.415 - 0.909i)T^{2} \)
47 \( 1 - 1.81T + T^{2} \)
53 \( 1 + (-0.841 + 0.540i)T^{2} \)
59 \( 1 + (-0.841 - 0.540i)T^{2} \)
61 \( 1 + (-0.239 - 0.153i)T + (0.415 + 0.909i)T^{2} \)
67 \( 1 + (0.708 - 0.817i)T + (-0.142 - 0.989i)T^{2} \)
71 \( 1 + (0.142 + 0.989i)T^{2} \)
73 \( 1 + (0.959 - 0.281i)T^{2} \)
79 \( 1 + (-0.841 - 0.540i)T^{2} \)
83 \( 1 + (0.449 - 0.983i)T + (-0.654 - 0.755i)T^{2} \)
89 \( 1 + (1.10 - 0.708i)T + (0.415 - 0.909i)T^{2} \)
97 \( 1 + (0.654 - 0.755i)T^{2} \)
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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

−9.664818417623991604996740741473, −9.102393776595161023767640860912, −8.178519681905919614793399409675, −7.17462736308209011337170663113, −6.23099717722052933928284002506, −5.50127732279376345528478399992, −4.83672101005339715288017509109, −4.13500363950947380310600543567, −3.09386655938849796228921046260, −1.18638156543007142538581776252, 0.46179538201442682120106401467, 2.09830792730490610564465291617, 3.27023232111438808093137298952, 4.31651081571803439537547471076, 5.47319054006383049613003095165, 6.13965139524490569941579063979, 6.94616900425860531895597955196, 7.21110273430466631598583575316, 8.232457501699396429875199219855, 9.298677403112354851410884792298

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