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Properties

Label 2-1859-11.10-c0-0-8
Degree $2$
Conductor $1859$
Sign $-i$
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  − 1.61i·2-s − 1.61·3-s − 1.61·4-s + 2.61i·6-s − 0.618i·7-s + i·8-s + 1.61·9-s − i·11-s + 2.61·12-s − 1.00·14-s − 2.61i·18-s − 1.61i·19-s + 1.00i·21-s − 1.61·22-s − 0.618·23-s − 1.61i·24-s + ⋯
L(s)  = 1  − 1.61i·2-s − 1.61·3-s − 1.61·4-s + 2.61i·6-s − 0.618i·7-s + i·8-s + 1.61·9-s − i·11-s + 2.61·12-s − 1.00·14-s − 2.61i·18-s − 1.61i·19-s + 1.00i·21-s − 1.61·22-s − 0.618·23-s − 1.61i·24-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $-i$
Analytic conductor: \(0.927761\)
Root analytic conductor: \(0.963203\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (846, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1859,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3148794867\)
\(L(\frac12)\) \(\approx\) \(0.3148794867\)
\(L(1)\) 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 \( 1 + 1.61iT - T^{2} \)
3 \( 1 + 1.61T + T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + 0.618iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 1.61iT - T^{2} \)
23 \( 1 + 0.618T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 0.618iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 1.61T + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - 1.61iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 0.618iT - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.319180596435391468905524411395, −8.242904528771363186590188274570, −7.07105492518959057742237749880, −6.29692062094946694710022732024, −5.39414066333506429847196884150, −4.55757157882547882637906045316, −3.84202967344876493561060808719, −2.71681443846355516991990598829, −1.32682616483191310955730332663, −0.31049275573800656140590215459, 1.86848457489357647895693523620, 3.97162095399606170653136423171, 4.79186006953995379447151004771, 5.53395571496240052042756517294, 6.02684996587430093022854228675, 6.60484534804742234873286411285, 7.52070925528654280645044196283, 8.039573512950707640090967953548, 9.162728367632730939685623980537, 9.912787891590282495378207967661

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