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Properties

Label 2-1840-460.399-c0-0-1
Degree $2$
Conductor $1840$
Sign $0.779 + 0.626i$
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  + (−0.822 − 1.80i)3-s + (0.654 + 0.755i)5-s + (0.909 + 0.584i)7-s + (−1.91 + 2.20i)9-s + (0.822 − 1.80i)15-s + (0.304 − 2.11i)21-s + (0.989 + 0.142i)23-s + (−0.142 + 0.989i)25-s + (3.64 + 1.07i)27-s + (1.25 − 0.368i)29-s + (0.153 + 1.07i)35-s + (1.10 + 1.27i)41-s + (−0.234 − 0.512i)43-s − 2.91·45-s − 1.51·47-s + ⋯
L(s)  = 1  + (−0.822 − 1.80i)3-s + (0.654 + 0.755i)5-s + (0.909 + 0.584i)7-s + (−1.91 + 2.20i)9-s + (0.822 − 1.80i)15-s + (0.304 − 2.11i)21-s + (0.989 + 0.142i)23-s + (−0.142 + 0.989i)25-s + (3.64 + 1.07i)27-s + (1.25 − 0.368i)29-s + (0.153 + 1.07i)35-s + (1.10 + 1.27i)41-s + (−0.234 − 0.512i)43-s − 2.91·45-s − 1.51·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.779 + 0.626i$
Analytic conductor: \(0.918279\)
Root analytic conductor: \(0.958269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :0),\ 0.779 + 0.626i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.053118455\)
\(L(\frac12)\) \(\approx\) \(1.053118455\)
\(L(1)\) 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 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (-0.989 - 0.142i)T \)
good3 \( 1 + (0.822 + 1.80i)T + (-0.654 + 0.755i)T^{2} \)
7 \( 1 + (-0.909 - 0.584i)T + (0.415 + 0.909i)T^{2} \)
11 \( 1 + (0.959 - 0.281i)T^{2} \)
13 \( 1 + (-0.415 + 0.909i)T^{2} \)
17 \( 1 + (-0.841 + 0.540i)T^{2} \)
19 \( 1 + (-0.841 - 0.540i)T^{2} \)
29 \( 1 + (-1.25 + 0.368i)T + (0.841 - 0.540i)T^{2} \)
31 \( 1 + (0.654 + 0.755i)T^{2} \)
37 \( 1 + (0.142 + 0.989i)T^{2} \)
41 \( 1 + (-1.10 - 1.27i)T + (-0.142 + 0.989i)T^{2} \)
43 \( 1 + (0.234 + 0.512i)T + (-0.654 + 0.755i)T^{2} \)
47 \( 1 + 1.51T + T^{2} \)
53 \( 1 + (-0.415 - 0.909i)T^{2} \)
59 \( 1 + (-0.415 + 0.909i)T^{2} \)
61 \( 1 + (-0.797 + 1.74i)T + (-0.654 - 0.755i)T^{2} \)
67 \( 1 + (-0.258 + 1.80i)T + (-0.959 - 0.281i)T^{2} \)
71 \( 1 + (0.959 + 0.281i)T^{2} \)
73 \( 1 + (-0.841 - 0.540i)T^{2} \)
79 \( 1 + (-0.415 + 0.909i)T^{2} \)
83 \( 1 + (1.19 - 1.37i)T + (-0.142 - 0.989i)T^{2} \)
89 \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \)
97 \( 1 + (0.142 - 0.989i)T^{2} \)
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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

−9.236853943375714677388401141860, −8.160986520672252969594802446945, −7.80543764084296498491718221892, −6.64838445398247445233824243029, −6.50780388156842258647322958562, −5.44947851288468000673011513075, −4.95312728390143487465532094050, −2.93152000439744461002475282391, −2.15322099495091424901022504415, −1.28341904220946221199591964011, 1.07532664774796184467286191395, 2.89111473859549983144773386814, 4.14379592528252453638562532936, 4.63107370414096145123138461781, 5.28402761856935828625820787616, 5.95071665157127494233404927627, 6.98277088485354329763795798756, 8.410325868396466483082526319760, 8.835394045376140919134943003258, 9.722918378995223244245209065633

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