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Properties

Label 2-1840-460.459-c1-0-70
Degree $2$
Conductor $1840$
Sign $0.147 + 0.989i$
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  + 3.16·3-s − 2.23i·5-s − 4.24i·7-s + 7.00·9-s − 7.07i·15-s − 13.4i·21-s + (−4.74 + 0.707i)23-s − 5.00·25-s + 12.6·27-s − 6·29-s − 9.48·35-s + 12·41-s + 12.7i·43-s − 15.6i·45-s + 9.48·47-s + ⋯
L(s)  = 1  + 1.82·3-s − 0.999i·5-s − 1.60i·7-s + 2.33·9-s − 1.82i·15-s − 2.92i·21-s + (−0.989 + 0.147i)23-s − 1.00·25-s + 2.43·27-s − 1.11·29-s − 1.60·35-s + 1.87·41-s + 1.94i·43-s − 2.33i·45-s + 1.38·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.147 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.147 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.147 + 0.989i$
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: \((2,\ 1840,\ (\ :1/2),\ 0.147 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.312105968\)
\(L(\frac12)\) \(\approx\) \(3.312105968\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23iT \)
23 \( 1 + (4.74 - 0.707i)T \)
good3 \( 1 - 3.16T + 3T^{2} \)
7 \( 1 + 4.24iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 - 12.7iT - 43T^{2} \)
47 \( 1 - 9.48T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 13.4iT - 61T^{2} \)
67 \( 1 + 4.24iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 15.5iT - 83T^{2} \)
89 \( 1 - 17.8iT - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−9.105697671966544422037816504857, −8.206154656231066002290505884728, −7.66981167545775576475447695974, −7.19058096652757856726012202484, −5.87828133552865796052554823830, −4.32511251589903727092955923506, −4.22353258160984330676185182317, −3.20672562077088951890911286649, −1.98267923644825071147193256303, −0.992719158316320777705041565287, 2.12864841305706741144718164091, 2.35789569964417640388409082685, 3.36165500083336533606413305160, 4.07293377587954730748701419088, 5.49144701961845764894916238254, 6.33830228616134470610451434397, 7.37557812733734881293790762204, 7.892227914853925266771239993353, 8.770174630115810353971532412091, 9.215403497678866252495955612694

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