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Properties

Label 2-1840-460.459-c1-0-66
Degree $2$
Conductor $1840$
Sign $-0.866 + 0.5i$
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  + 2.23·5-s − 4.33i·7-s − 3·9-s − 7.50·17-s − 4.79i·23-s + 5.00·25-s − 8.78·29-s − 2.76i·31-s − 9.69i·35-s + 1.43·37-s − 12.7·41-s + 9.59i·43-s − 6.70·45-s − 11.7·49-s + 13.5·53-s + ⋯
L(s)  = 1  + 0.999·5-s − 1.63i·7-s − 9-s − 1.82·17-s − 0.999i·23-s + 1.00·25-s − 1.63·29-s − 0.496i·31-s − 1.63i·35-s + 0.236·37-s − 1.99·41-s + 1.46i·43-s − 0.999·45-s − 1.68·49-s + 1.86·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.866 + 0.5i$
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: \((2,\ 1840,\ (\ :1/2),\ -0.866 + 0.5i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9403088643\)
\(L(\frac12)\) \(\approx\) \(0.9403088643\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23T \)
23 \( 1 + 4.79iT \)
good3 \( 1 + 3T^{2} \)
7 \( 1 + 4.33iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 7.50T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
29 \( 1 + 8.78T + 29T^{2} \)
31 \( 1 + 2.76iT - 31T^{2} \)
37 \( 1 - 1.43T + 37T^{2} \)
41 \( 1 + 12.7T + 41T^{2} \)
43 \( 1 - 9.59iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 13.5T + 53T^{2} \)
59 \( 1 - 4.16iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 11.1iT - 67T^{2} \)
71 \( 1 + 16.6iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 2.48iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 4.47T + 97T^{2} \)
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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

−8.948219614264404708248828258699, −8.245110308070158140317053705378, −7.15528920482051563327362715960, −6.60122614804878787742450104004, −5.78863565717786573961320839524, −4.77141480057079528305573809145, −3.99344465131976454224075449409, −2.81794666121454283782123122131, −1.78237309686091182520245304873, −0.30772117322514140432805776825, 1.96457940962829690298066018271, 2.43061103322267684309924188724, 3.55728118899011783371262592912, 5.12080678932262965741698592465, 5.48814767882013500061529269119, 6.24140790015512100619186057071, 7.02133594978100657788180939045, 8.367101213699568166798683599962, 8.932287119261729372772344544319, 9.254922261210134813684190918475

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