Properties

Label 2-1840-5.4-c1-0-49
Degree $2$
Conductor $1840$
Sign $-0.321 + 0.946i$
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  − 2.49i·3-s + (2.11 + 0.719i)5-s − 2.92i·7-s − 3.24·9-s + 4.10·11-s + 0.0122i·13-s + (1.79 − 5.29i)15-s + 0.155i·17-s + 4.32·19-s − 7.31·21-s i·23-s + (3.96 + 3.04i)25-s + 0.616i·27-s + 6.79·29-s − 2.20·31-s + ⋯
L(s)  = 1  − 1.44i·3-s + (0.946 + 0.321i)5-s − 1.10i·7-s − 1.08·9-s + 1.23·11-s + 0.00341i·13-s + (0.464 − 1.36i)15-s + 0.0376i·17-s + 0.992·19-s − 1.59·21-s − 0.208i·23-s + (0.792 + 0.609i)25-s + 0.118i·27-s + 1.26·29-s − 0.396·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.321 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.321 + 0.946i$
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: \((2,\ 1840,\ (\ :1/2),\ -0.321 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.307609744\)
\(L(\frac12)\) \(\approx\) \(2.307609744\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.11 - 0.719i)T \)
23 \( 1 + iT \)
good3 \( 1 + 2.49iT - 3T^{2} \)
7 \( 1 + 2.92iT - 7T^{2} \)
11 \( 1 - 4.10T + 11T^{2} \)
13 \( 1 - 0.0122iT - 13T^{2} \)
17 \( 1 - 0.155iT - 17T^{2} \)
19 \( 1 - 4.32T + 19T^{2} \)
29 \( 1 - 6.79T + 29T^{2} \)
31 \( 1 + 2.20T + 31T^{2} \)
37 \( 1 + 4.60iT - 37T^{2} \)
41 \( 1 + 7.67T + 41T^{2} \)
43 \( 1 - 8.38iT - 43T^{2} \)
47 \( 1 + 6.38iT - 47T^{2} \)
53 \( 1 - 7.80iT - 53T^{2} \)
59 \( 1 + 14.1T + 59T^{2} \)
61 \( 1 - 7.05T + 61T^{2} \)
67 \( 1 + 7.31iT - 67T^{2} \)
71 \( 1 - 5.84T + 71T^{2} \)
73 \( 1 + 0.727iT - 73T^{2} \)
79 \( 1 - 4.81T + 79T^{2} \)
83 \( 1 - 7.75iT - 83T^{2} \)
89 \( 1 + 6.77T + 89T^{2} \)
97 \( 1 - 14.8iT - 97T^{2} \)
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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

−9.013822719979554005894528378954, −7.999243755351663052762772588850, −7.25537615855980897165344230267, −6.65464105358477615280809120143, −6.22438270481089772235913826744, −5.09812057496902495864886335490, −3.88417244843440239846830286540, −2.79383975557866943112699659306, −1.61313879119983150664295349983, −0.976671932290797102505285524924, 1.49131794945646159828252517923, 2.77306616412455137762758904837, 3.67530849471521027947222615970, 4.73848571271777618734400166954, 5.30524273127363191521616185512, 6.04856867430268572334858867515, 6.90010112802740712827128938810, 8.402226018680356251301665889444, 8.936958828682424025197633583644, 9.535182536090068182668503643669

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