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Properties

Label 2-1840-460.239-c0-0-0
Degree $2$
Conductor $1840$
Sign $-0.394 - 0.918i$
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  + (0.368 + 0.425i)3-s + (0.142 + 0.989i)5-s + (−0.755 + 1.65i)7-s + (0.0971 − 0.675i)9-s + (−0.368 + 0.425i)15-s + (−0.983 + 0.288i)21-s + (0.281 + 0.959i)23-s + (−0.959 + 0.281i)25-s + (0.797 − 0.512i)27-s + (−0.239 − 0.153i)29-s + (−1.74 − 0.512i)35-s + (0.118 + 0.822i)41-s + (0.708 + 0.817i)43-s + 0.682·45-s − 1.97·47-s + ⋯
L(s)  = 1  + (0.368 + 0.425i)3-s + (0.142 + 0.989i)5-s + (−0.755 + 1.65i)7-s + (0.0971 − 0.675i)9-s + (−0.368 + 0.425i)15-s + (−0.983 + 0.288i)21-s + (0.281 + 0.959i)23-s + (−0.959 + 0.281i)25-s + (0.797 − 0.512i)27-s + (−0.239 − 0.153i)29-s + (−1.74 − 0.512i)35-s + (0.118 + 0.822i)41-s + (0.708 + 0.817i)43-s + 0.682·45-s − 1.97·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.394 - 0.918i$
Analytic conductor: \(0.918279\)
Root analytic conductor: \(0.958269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :0),\ -0.394 - 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.134136178\)
\(L(\frac12)\) \(\approx\) \(1.134136178\)
\(L(1)\) 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 \( 1 + (-0.142 - 0.989i)T \)
23 \( 1 + (-0.281 - 0.959i)T \)
good3 \( 1 + (-0.368 - 0.425i)T + (-0.142 + 0.989i)T^{2} \)
7 \( 1 + (0.755 - 1.65i)T + (-0.654 - 0.755i)T^{2} \)
11 \( 1 + (-0.841 - 0.540i)T^{2} \)
13 \( 1 + (0.654 - 0.755i)T^{2} \)
17 \( 1 + (-0.415 - 0.909i)T^{2} \)
19 \( 1 + (-0.415 + 0.909i)T^{2} \)
29 \( 1 + (0.239 + 0.153i)T + (0.415 + 0.909i)T^{2} \)
31 \( 1 + (0.142 + 0.989i)T^{2} \)
37 \( 1 + (0.959 + 0.281i)T^{2} \)
41 \( 1 + (-0.118 - 0.822i)T + (-0.959 + 0.281i)T^{2} \)
43 \( 1 + (-0.708 - 0.817i)T + (-0.142 + 0.989i)T^{2} \)
47 \( 1 + 1.97T + T^{2} \)
53 \( 1 + (0.654 + 0.755i)T^{2} \)
59 \( 1 + (0.654 - 0.755i)T^{2} \)
61 \( 1 + (-1.10 + 1.27i)T + (-0.142 - 0.989i)T^{2} \)
67 \( 1 + (1.45 - 0.425i)T + (0.841 - 0.540i)T^{2} \)
71 \( 1 + (-0.841 + 0.540i)T^{2} \)
73 \( 1 + (-0.415 + 0.909i)T^{2} \)
79 \( 1 + (0.654 - 0.755i)T^{2} \)
83 \( 1 + (-0.215 + 1.49i)T + (-0.959 - 0.281i)T^{2} \)
89 \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \)
97 \( 1 + (0.959 - 0.281i)T^{2} \)
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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

−9.622884724189096992273836417323, −9.113616421640308653336459627588, −8.281654823437477102932476162243, −7.25881208343963790104250460147, −6.27895850458588596782871492747, −5.97409383065353476489264334566, −4.82339160126138421484413947235, −3.43767824944645185418180576849, −3.08390670842130696582390605450, −2.04917417356493426395222551226, 0.809994956083620348411145042731, 2.00114333289604169111408416998, 3.32037644538204256481626619195, 4.27450448825052414771172450144, 4.94852111886658032498214042375, 6.09777287273895059006414257137, 7.03999022991823787734888053762, 7.55512420670341198441025136882, 8.389757742583893545147265495466, 9.095213812156533493276482946064

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