Properties

Label 2-1840-5.4-c1-0-36
Degree $2$
Conductor $1840$
Sign $0.447 - 0.894i$
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  + 2i·3-s + (2 + i)5-s i·7-s − 9-s − 2i·13-s + (−2 + 4i)15-s − 5i·17-s + 8·19-s + 2·21-s i·23-s + (3 + 4i)25-s + 4i·27-s + 5·29-s + 5·31-s + (1 − 2i)35-s + ⋯
L(s)  = 1  + 1.15i·3-s + (0.894 + 0.447i)5-s − 0.377i·7-s − 0.333·9-s − 0.554i·13-s + (−0.516 + 1.03i)15-s − 1.21i·17-s + 1.83·19-s + 0.436·21-s − 0.208i·23-s + (0.600 + 0.800i)25-s + 0.769i·27-s + 0.928·29-s + 0.898·31-s + (0.169 − 0.338i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(14.6924\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.272261699\)
\(L(\frac12)\) \(\approx\) \(2.272261699\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - i)T \)
23 \( 1 + iT \)
good3 \( 1 - 2iT - 3T^{2} \)
7 \( 1 + iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 5iT - 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 - 7iT - 37T^{2} \)
41 \( 1 + 7T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 - iT - 53T^{2} \)
59 \( 1 - 3T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + 13iT - 67T^{2} \)
71 \( 1 + 13T + 71T^{2} \)
73 \( 1 + 8iT - 73T^{2} \)
79 \( 1 + 14T + 79T^{2} \)
83 \( 1 + 3iT - 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−9.576148030410108098600644602424, −8.937412046044829089611044227641, −7.76313551498356507799038903369, −7.00935219860697231240192946526, −6.09269721317024906131007890852, −5.09448648118850537508863782361, −4.70235207395148745835275342817, −3.32978838267781671944453397696, −2.84274349368173337774974213154, −1.17306808228243964742902952309, 1.07977648410735711665803284198, 1.83492881296117993969960863091, 2.81783495171724879901785858667, 4.18722899653004125931631119656, 5.33328016647002968014205843757, 5.95137489070646669466531836892, 6.74325802401862703672458195401, 7.43975023962131211940246693187, 8.375765579340527809822076059588, 8.951532487729763002332319976483

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