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Properties

Label 2-1911-13.12-c1-0-49
Degree $2$
Conductor $1911$
Sign $0.277 + 0.960i$
Analytic cond. $15.2594$
Root an. cond. $3.90632$
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  − 1.73i·2-s + 3-s − 0.999·4-s − 1.73i·6-s − 1.73i·8-s + 9-s + 3.46i·11-s − 0.999·12-s + (1 + 3.46i)13-s − 5·16-s + 6·17-s − 1.73i·18-s + 3.46i·19-s + 5.99·22-s − 1.73i·24-s + 5·25-s + ⋯
L(s)  = 1  − 1.22i·2-s + 0.577·3-s − 0.499·4-s − 0.707i·6-s − 0.612i·8-s + 0.333·9-s + 1.04i·11-s − 0.288·12-s + (0.277 + 0.960i)13-s − 1.25·16-s + 1.45·17-s − 0.408i·18-s + 0.794i·19-s + 1.27·22-s − 0.353i·24-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.277 + 0.960i$
Analytic conductor: \(15.2594\)
Root analytic conductor: \(3.90632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :1/2),\ 0.277 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.476076718\)
\(L(\frac12)\) \(\approx\) \(2.476076718\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 \)
13 \( 1 + (-1 - 3.46i)T \)
good2 \( 1 + 1.73iT - 2T^{2} \)
5 \( 1 - 5T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + 6.92iT - 37T^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 10.3iT - 67T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 + 13.8iT - 97T^{2} \)
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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

−9.245000306680242023247519078622, −8.539289115386261900455166692662, −7.43951926691108132379489700808, −6.91653241907340992493790574238, −5.76959299417584531683090667326, −4.51376298170651482452160570647, −3.87927765565575955518738992643, −2.95739467918423880837004008699, −2.05124797008738803673214720642, −1.18115291536560280564072047293, 1.04718261918359056732920051980, 2.78230233633233724270467661169, 3.37272683709823237100446743701, 4.84513511667225839469005802660, 5.42724079160671554520722762783, 6.36800128482475557087438731826, 6.94899996177830681580331352134, 8.060594527171942642100862064384, 8.197768146782453502307219552630, 8.985153032263949423271726122210

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