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Properties

Label 2-1859-11.10-c0-0-9
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  − 0.618i·2-s + 0.618·3-s + 0.618·4-s − 0.381i·6-s − 1.61i·7-s − i·8-s − 0.618·9-s + i·11-s + 0.381·12-s − 1.00·14-s + 0.381i·18-s − 0.618i·19-s − 1.00i·21-s + 0.618·22-s + 1.61·23-s − 0.618i·24-s + ⋯
L(s)  = 1  − 0.618i·2-s + 0.618·3-s + 0.618·4-s − 0.381i·6-s − 1.61i·7-s − i·8-s − 0.618·9-s + i·11-s + 0.381·12-s − 1.00·14-s + 0.381i·18-s − 0.618i·19-s − 1.00i·21-s + 0.618·22-s + 1.61·23-s − 0.618i·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

\(L(\frac{1}{2})\) \(\approx\) \(1.586387097\)
\(L(\frac12)\) \(\approx\) \(1.586387097\)
\(L(1)\) 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 \( 1 + 0.618iT - T^{2} \)
3 \( 1 - 0.618T + T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + 1.61iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 0.618iT - T^{2} \)
23 \( 1 - 1.61T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 1.61iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - 0.618T + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - 0.618iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.61iT - T^{2} \)
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

−9.560974235431865695038951133234, −8.437866300885635453894284120157, −7.45592176173952934808443526336, −7.14936399609368265597987068168, −6.25646414203093292681187990034, −4.90487023365986213254722904610, −3.99550743835623145647912398860, −3.21031578507652648803809475803, −2.32744956284831945345976316818, −1.14676397370918974089592620189, 1.97554491217688452470336438071, 2.75936600320791000665323816245, 3.48053015922686279154758778667, 5.19160015289070285605485097175, 5.74092703204870365036091661391, 6.27297140662719251114481611182, 7.38594170513729634414646024494, 8.137210326517093829113721998144, 8.814435850198006607853783526536, 9.095925369551746139538820557797

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