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Properties

Label 2-3800-5.4-c1-0-63
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  − 2i·3-s + 3i·7-s − 9-s − 3·11-s − 4i·13-s − 5i·17-s + 19-s + 6·21-s − 4i·27-s − 2·29-s + 8·31-s + 6i·33-s + 10i·37-s − 8·39-s + 6·41-s + ⋯
L(s)  = 1  − 1.15i·3-s + 1.13i·7-s − 0.333·9-s − 0.904·11-s − 1.10i·13-s − 1.21i·17-s + 0.229·19-s + 1.30·21-s − 0.769i·27-s − 0.371·29-s + 1.43·31-s + 1.04i·33-s + 1.64i·37-s − 1.28·39-s + 0.937·41-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

\(L(1)\) \(\approx\) \(1.178716505\)
\(L(\frac12)\) \(\approx\) \(1.178716505\)
\(L(\frac{3}{2})\) 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 \( 1 + 2iT - 3T^{2} \)
7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + 5iT - 17T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 7iT - 43T^{2} \)
47 \( 1 - 9iT - 47T^{2} \)
53 \( 1 + 8iT - 53T^{2} \)
59 \( 1 + 14T + 59T^{2} \)
61 \( 1 + 5T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 15iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 16iT - 97T^{2} \)
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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

−7.904255794823810826081156019125, −7.68841256722481721106082275195, −6.71048135358289171715399657223, −6.00369101759861900478797809969, −5.33072368917676010425119398480, −4.59814527797673581780685692576, −2.90207909485338735430160526495, −2.73670327717184577594268208890, −1.53455697870083403354735221838, −0.35376852483011799093684954549, 1.28912794278658653083113113004, 2.56870775710239608364879151050, 3.73458939752118664797158719667, 4.17522762493071544664550711140, 4.79527442298231450527328722980, 5.73860933803634296000800871830, 6.59216034850201340661285811958, 7.44808859959028274199562971376, 8.025698111366362996554971981534, 9.030809037733862156623656225545

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