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Properties

Label 2-3800-5.4-c1-0-79
Degree $2$
Conductor $3800$
Sign $-0.894 - 0.447i$
Analytic cond. $30.3431$
Root an. cond. $5.50846$
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  − 1.76i·3-s − 4.62i·7-s − 0.103·9-s − 5.52·11-s − 5.49i·13-s − 6.62i·17-s + 19-s − 8.14·21-s − 4.14i·23-s − 5.10i·27-s + 7.87·29-s + 1.25·31-s + 9.72i·33-s − 0.387i·37-s − 9.67·39-s + ⋯
L(s)  = 1  − 1.01i·3-s − 1.74i·7-s − 0.0343·9-s − 1.66·11-s − 1.52i·13-s − 1.60i·17-s + 0.229·19-s − 1.77·21-s − 0.865i·23-s − 0.982i·27-s + 1.46·29-s + 0.224·31-s + 1.69i·33-s − 0.0637i·37-s − 1.54·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(30.3431\)
Root analytic conductor: \(5.50846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (3649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.530556371\)
\(L(\frac12)\) \(\approx\) \(1.530556371\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 - T \)
good3 \( 1 + 1.76iT - 3T^{2} \)
7 \( 1 + 4.62iT - 7T^{2} \)
11 \( 1 + 5.52T + 11T^{2} \)
13 \( 1 + 5.49iT - 13T^{2} \)
17 \( 1 + 6.62iT - 17T^{2} \)
23 \( 1 + 4.14iT - 23T^{2} \)
29 \( 1 - 7.87T + 29T^{2} \)
31 \( 1 - 1.25T + 31T^{2} \)
37 \( 1 + 0.387iT - 37T^{2} \)
41 \( 1 - 6.77T + 41T^{2} \)
43 \( 1 - 10.9iT - 43T^{2} \)
47 \( 1 - 1.72iT - 47T^{2} \)
53 \( 1 + 1.49iT - 53T^{2} \)
59 \( 1 + 0.626T + 59T^{2} \)
61 \( 1 - 15.0T + 61T^{2} \)
67 \( 1 - 5.22iT - 67T^{2} \)
71 \( 1 + 11.0T + 71T^{2} \)
73 \( 1 - 4.83iT - 73T^{2} \)
79 \( 1 - 2.98T + 79T^{2} \)
83 \( 1 - 2.74iT - 83T^{2} \)
89 \( 1 + 4.27T + 89T^{2} \)
97 \( 1 + 13.7iT - 97T^{2} \)
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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

−7.889893680909231456890231369774, −7.36718153166328054233246870416, −6.91259796970260692414781318447, −5.95484426972351864196793547063, −4.97342780122875455047696196600, −4.40923591793889645791041773897, −3.05364215897853056653497623342, −2.56754261758885739691919033778, −0.949178259634359153352993798689, −0.54236866150373580835468853725, 1.82820213190766681649474878966, 2.55965775432027961233084766395, 3.55775146505649939942628148591, 4.45010469299731146659966239112, 5.18080520841151886140763018977, 5.73310335969911330593803927912, 6.53895679688382680505264045220, 7.58999428143204628278591229663, 8.479531534791128939491059187719, 8.870331992407754769187663822878

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