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Properties

Label 2-1840-460.459-c1-0-65
Degree $2$
Conductor $1840$
Sign $-0.927 - 0.373i$
Analytic cond. $14.6924$
Root an. cond. $3.83307$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.924·3-s + (0.611 − 2.15i)5-s − 1.51i·7-s − 2.14·9-s − 0.770·11-s − 4.68i·13-s + (−0.564 + 1.98i)15-s − 4.00·17-s + 7.17·19-s + 1.39i·21-s + (−4.38 + 1.94i)23-s + (−4.25 − 2.62i)25-s + 4.75·27-s − 4.41·29-s + 3.07i·31-s + ⋯
L(s)  = 1  − 0.533·3-s + (0.273 − 0.961i)5-s − 0.572i·7-s − 0.715·9-s − 0.232·11-s − 1.29i·13-s + (−0.145 + 0.513i)15-s − 0.972·17-s + 1.64·19-s + 0.305i·21-s + (−0.914 + 0.405i)23-s + (−0.850 − 0.525i)25-s + 0.915·27-s − 0.820·29-s + 0.551i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.927 - 0.373i$
Analytic conductor: \(14.6924\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1839, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ -0.927 - 0.373i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3835348043\)
\(L(\frac12)\) \(\approx\) \(0.3835348043\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.611 + 2.15i)T \)
23 \( 1 + (4.38 - 1.94i)T \)
good3 \( 1 + 0.924T + 3T^{2} \)
7 \( 1 + 1.51iT - 7T^{2} \)
11 \( 1 + 0.770T + 11T^{2} \)
13 \( 1 + 4.68iT - 13T^{2} \)
17 \( 1 + 4.00T + 17T^{2} \)
19 \( 1 - 7.17T + 19T^{2} \)
29 \( 1 + 4.41T + 29T^{2} \)
31 \( 1 - 3.07iT - 31T^{2} \)
37 \( 1 + 5.52T + 37T^{2} \)
41 \( 1 - 7.75T + 41T^{2} \)
43 \( 1 - 2.03iT - 43T^{2} \)
47 \( 1 + 4.08T + 47T^{2} \)
53 \( 1 + 10.2T + 53T^{2} \)
59 \( 1 + 0.831iT - 59T^{2} \)
61 \( 1 - 7.18iT - 61T^{2} \)
67 \( 1 - 12.6iT - 67T^{2} \)
71 \( 1 + 2.23iT - 71T^{2} \)
73 \( 1 + 11.1iT - 73T^{2} \)
79 \( 1 + 3.98T + 79T^{2} \)
83 \( 1 - 7.33iT - 83T^{2} \)
89 \( 1 - 1.08iT - 89T^{2} \)
97 \( 1 - 2.74T + 97T^{2} \)
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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

−8.823670823064415559897161131189, −8.002822665079926154380083193985, −7.36978197889024339756337322382, −6.16432426552830573866199409042, −5.48893290755597589838258727028, −4.96892880650955415713222518251, −3.84479673372016458188183568626, −2.76568932274160162318454972980, −1.29791349341956726154309534704, −0.15303778286620066280075512439, 1.92276453694063841702601746702, 2.75977766608583366073500309628, 3.82605328227737369125068386384, 4.98811812254732580799204908145, 5.82035704555114671856083049394, 6.40496527847161208935033218585, 7.17709182851507091742063674308, 8.062808590942303098616599136441, 9.128933275171510437093448140639, 9.558423062607592067032931326839

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