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Properties

Label 2-1859-143.10-c0-0-5
Degree $2$
Conductor $1859$
Sign $0.993 - 0.114i$
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  + (0.900 + 1.56i)3-s + (0.5 − 0.866i)4-s − 1.24i·5-s + (−1.12 + 1.94i)9-s + (0.866 − 0.5i)11-s + 1.80·12-s + (1.94 − 1.12i)15-s + (−0.499 − 0.866i)16-s + (−1.07 − 0.623i)20-s + (−0.222 − 0.385i)23-s − 0.554·25-s − 2.24·27-s + 0.445i·31-s + (1.56 + 0.900i)33-s + (1.12 + 1.94i)36-s + (−1.56 + 0.900i)37-s + ⋯
L(s)  = 1  + (0.900 + 1.56i)3-s + (0.5 − 0.866i)4-s − 1.24i·5-s + (−1.12 + 1.94i)9-s + (0.866 − 0.5i)11-s + 1.80·12-s + (1.94 − 1.12i)15-s + (−0.499 − 0.866i)16-s + (−1.07 − 0.623i)20-s + (−0.222 − 0.385i)23-s − 0.554·25-s − 2.24·27-s + 0.445i·31-s + (1.56 + 0.900i)33-s + (1.12 + 1.94i)36-s + (−1.56 + 0.900i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $0.993 - 0.114i$
Analytic conductor: \(0.927761\)
Root analytic conductor: \(0.963203\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (868, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1859,\ (\ :0),\ 0.993 - 0.114i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.731069049\)
\(L(\frac12)\) \(\approx\) \(1.731069049\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + 1.24iT - T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - 0.445iT - T^{2} \)
37 \( 1 + (1.56 - 0.900i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + 0.445iT - T^{2} \)
53 \( 1 - 1.24T + T^{2} \)
59 \( 1 + (-1.56 - 0.900i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (1.56 + 0.900i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.385 - 0.222i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-1.07 - 0.623i)T + (0.5 + 0.866i)T^{2} \)
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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

−9.290860260451263871162974928578, −8.861638394952980644118418351201, −8.378855207525571699713381435375, −7.12525659889477016770534132686, −5.93872723691288035469685854388, −5.20142815204592756575182593614, −4.53962072387273503791545150532, −3.74853876978444209471220802541, −2.65946340404853961838069621874, −1.37506547057488296015897975529, 1.73069473338422020435223031279, 2.43792290516897215234245897445, 3.26517680831476565795269157356, 3.94502850243591088059100189237, 5.88617167854008886709773230430, 6.73762110286945278164160016330, 7.09855322318757098533061646599, 7.56739799407674603000251838800, 8.463177915444221192388356674481, 9.067179740179071259738412059381

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