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Properties

Label 2-1840-460.219-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: $\chi_{1840} (1599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :0),\ -0.320 + 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.277295855\)
\(L(\frac12)\) \(\approx\) \(1.277295855\)
\(L(1)\) 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

−9.510050811679417166504635695534, −8.202129347206729552792592040463, −7.58625336509898968117259837596, −6.66092022582596609068155523984, −6.39151092401094530674636859296, −5.29478984869608291444356212624, −4.18133603434797674723010347522, −3.10525283938713685138350798467, −1.98918162592237202362650938239, −0.971254721685070402317675617533, 1.96121763690997354116373225554, 3.01056238020387829426521311514, 3.65063675794668156520425247782, 5.04207131442443271005814700038, 5.44624260034562421010173223222, 6.49544175421944129493320677283, 7.02187270974985577459881712585, 8.610184758386035532441791083161, 9.041504028497365764953209481815, 9.619443407228338329502943660093

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