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Properties

Label 2-1859-143.142-c0-0-0
Degree $2$
Conductor $1859$
Sign $-0.969 + 0.246i$
Analytic cond. $0.927761$
Root an. cond. $0.963203$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.445·3-s − 4-s + 1.80i·5-s − 0.801·9-s + i·11-s + 0.445·12-s − 0.801i·15-s + 16-s − 1.80i·20-s − 1.24·23-s − 2.24·25-s + 0.801·27-s − 1.24i·31-s − 0.445i·33-s + 0.801·36-s − 0.445i·37-s + ⋯
L(s)  = 1  − 0.445·3-s − 4-s + 1.80i·5-s − 0.801·9-s + i·11-s + 0.445·12-s − 0.801i·15-s + 16-s − 1.80i·20-s − 1.24·23-s − 2.24·25-s + 0.801·27-s − 1.24i·31-s − 0.445i·33-s + 0.801·36-s − 0.445i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $-0.969 + 0.246i$
Analytic conductor: \(0.927761\)
Root analytic conductor: \(0.963203\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (1858, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1859,\ (\ :0),\ -0.969 + 0.246i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2930224910\)
\(L(\frac12)\) \(\approx\) \(0.2930224910\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 - iT \)
13 \( 1 \)
good2 \( 1 + T^{2} \)
3 \( 1 + 0.445T + T^{2} \)
5 \( 1 - 1.80iT - T^{2} \)
7 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + 1.24T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.24iT - T^{2} \)
37 \( 1 + 0.445iT - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 1.24iT - T^{2} \)
53 \( 1 + 1.80T + T^{2} \)
59 \( 1 + 0.445iT - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 2iT - T^{2} \)
71 \( 1 - 0.445iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - 1.24iT - T^{2} \)
97 \( 1 - 1.80iT - T^{2} \)
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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

−9.791856218928602475220691120748, −9.372964704282645889858320035034, −8.022178463324399629065914330576, −7.64870284373954334731869563123, −6.46522097626899650852636561172, −6.05402286874379414525439666504, −5.00510530999665855025943364525, −4.03479916781289532546382655910, −3.16984102002718228387571426221, −2.14178456393599337502771814203, 0.24245434844708582607364550312, 1.42997438136705833995985811286, 3.26242389319779487910089372034, 4.25575753743660300192435730888, 5.02870039478073668893071613770, 5.55733381611124639299064317005, 6.25441069216201626622221367752, 7.83957302958750188980174574122, 8.582488954589370774741693515391, 8.692920536384541499587842658846

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