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Properties

Label 2-1840-5.4-c1-0-59
Degree $2$
Conductor $1840$
Sign $-0.992 - 0.122i$
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  − 2.98i·3-s + (0.274 − 2.21i)5-s − 0.980i·7-s − 5.92·9-s + 6.14·11-s − 6.37i·13-s + (−6.62 − 0.819i)15-s − 3.36i·17-s + 1.08·19-s − 2.92·21-s + i·23-s + (−4.84 − 1.21i)25-s + 8.72i·27-s + 0.271·29-s + 8.77·31-s + ⋯
L(s)  = 1  − 1.72i·3-s + (0.122 − 0.992i)5-s − 0.370i·7-s − 1.97·9-s + 1.85·11-s − 1.76i·13-s + (−1.71 − 0.211i)15-s − 0.815i·17-s + 0.248·19-s − 0.639·21-s + 0.208i·23-s + (−0.969 − 0.243i)25-s + 1.68i·27-s + 0.0503·29-s + 1.57·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.992 - 0.122i$
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: \((2,\ 1840,\ (\ :1/2),\ -0.992 - 0.122i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.896615898\)
\(L(\frac12)\) \(\approx\) \(1.896615898\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.274 + 2.21i)T \)
23 \( 1 - iT \)
good3 \( 1 + 2.98iT - 3T^{2} \)
7 \( 1 + 0.980iT - 7T^{2} \)
11 \( 1 - 6.14T + 11T^{2} \)
13 \( 1 + 6.37iT - 13T^{2} \)
17 \( 1 + 3.36iT - 17T^{2} \)
19 \( 1 - 1.08T + 19T^{2} \)
29 \( 1 - 0.271T + 29T^{2} \)
31 \( 1 - 8.77T + 31T^{2} \)
37 \( 1 - 8.84iT - 37T^{2} \)
41 \( 1 + 4.85T + 41T^{2} \)
43 \( 1 + 1.87iT - 43T^{2} \)
47 \( 1 - 0.196iT - 47T^{2} \)
53 \( 1 + 1.93iT - 53T^{2} \)
59 \( 1 - 13.0T + 59T^{2} \)
61 \( 1 - 6.00T + 61T^{2} \)
67 \( 1 - 2.26iT - 67T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 - 1.38iT - 73T^{2} \)
79 \( 1 + 4.67T + 79T^{2} \)
83 \( 1 - 15.7iT - 83T^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 - 10.2iT - 97T^{2} \)
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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

−8.489210706027525236275862831201, −8.149432062257481292962954544678, −7.20852682662717939079904732148, −6.58452719380774232772292387146, −5.79258686722512564670709340264, −4.94703929014686042301658240436, −3.67854842897374404476525413949, −2.54472415112629570515935724888, −1.18036471951747857334829110815, −0.847875693579928602973065173965, 1.91321283659269739923842395608, 3.14180892093613197759683964940, 4.09224115550871302580844155270, 4.30308976995122437443046078133, 5.67587331627675199284241854055, 6.39411972149559986212657217342, 7.04427788389456437924460004744, 8.541730676660920353218372688023, 9.043962704024480802930116025234, 9.687342816816596910900396234364

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