Properties

Label 2-1840-92.91-c1-0-9
Degree $2$
Conductor $1840$
Sign $0.317 - 0.948i$
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  − 0.657i·3-s i·5-s − 1.07·7-s + 2.56·9-s − 5.98·11-s − 3.56·13-s − 0.657·15-s + 4.56i·17-s + 4.31·19-s + 0.706i·21-s + (−3.59 + 3.17i)23-s − 25-s − 3.66i·27-s + 9.42·29-s + 9.28i·31-s + ⋯
L(s)  = 1  − 0.379i·3-s − 0.447i·5-s − 0.406·7-s + 0.855·9-s − 1.80·11-s − 0.989·13-s − 0.169·15-s + 1.10i·17-s + 0.990·19-s + 0.154i·21-s + (−0.749 + 0.662i)23-s − 0.200·25-s − 0.704i·27-s + 1.75·29-s + 1.66i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.317 - 0.948i$
Analytic conductor: \(14.6924\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ 0.317 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9658626813\)
\(L(\frac12)\) \(\approx\) \(0.9658626813\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
23 \( 1 + (3.59 - 3.17i)T \)
good3 \( 1 + 0.657iT - 3T^{2} \)
7 \( 1 + 1.07T + 7T^{2} \)
11 \( 1 + 5.98T + 11T^{2} \)
13 \( 1 + 3.56T + 13T^{2} \)
17 \( 1 - 4.56iT - 17T^{2} \)
19 \( 1 - 4.31T + 19T^{2} \)
29 \( 1 - 9.42T + 29T^{2} \)
31 \( 1 - 9.28iT - 31T^{2} \)
37 \( 1 - 8.93iT - 37T^{2} \)
41 \( 1 - 0.656T + 41T^{2} \)
43 \( 1 - 2.14T + 43T^{2} \)
47 \( 1 - 1.86iT - 47T^{2} \)
53 \( 1 + 1.79iT - 53T^{2} \)
59 \( 1 + 3.68iT - 59T^{2} \)
61 \( 1 + 4.08iT - 61T^{2} \)
67 \( 1 - 3.50T + 67T^{2} \)
71 \( 1 - 3.90iT - 71T^{2} \)
73 \( 1 - 12.6T + 73T^{2} \)
79 \( 1 + 7.92T + 79T^{2} \)
83 \( 1 + 3.50T + 83T^{2} \)
89 \( 1 + 5.13iT - 89T^{2} \)
97 \( 1 - 15.7iT - 97T^{2} \)
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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

−9.650007077573822694298673366869, −8.319904862409869760293830361364, −7.938720409372996422800905795169, −7.10831615108866090626401038693, −6.30071044481512643331956228728, −5.19371613850318302518695624408, −4.73010399395105040924123637287, −3.41986247055827337990624262717, −2.44292772312048892734921489727, −1.24436456657681677008963060912, 0.36958709871258417352989823337, 2.37631553403519783934550688767, 2.93565917850420301216990170452, 4.21709341950376368657528577756, 4.98287745169782553163150574146, 5.74947754278447892215076511356, 6.91338025143156111379030419148, 7.49349269778544870255590381822, 8.114949638768807524420203984526, 9.444904339894047505710086567746

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