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Properties

Label 2-3800-5.4-c1-0-36
Degree $2$
Conductor $3800$
Sign $0.447 - 0.894i$
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  − 0.185i·3-s + 4.45i·7-s + 2.96·9-s + 2.64·11-s + 1.30i·13-s + 3.51i·17-s + 19-s + 0.826·21-s − 6.52i·23-s − 1.10i·27-s + 5.20·29-s + 10.8·31-s − 0.490i·33-s − 2.04i·37-s + 0.241·39-s + ⋯
L(s)  = 1  − 0.107i·3-s + 1.68i·7-s + 0.988·9-s + 0.797·11-s + 0.360i·13-s + 0.853i·17-s + 0.229·19-s + 0.180·21-s − 1.36i·23-s − 0.212i·27-s + 0.967·29-s + 1.94·31-s − 0.0854i·33-s − 0.336i·37-s + 0.0386·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.447 - 0.894i$
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: \((2,\ 3800,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.265602260\)
\(L(\frac12)\) \(\approx\) \(2.265602260\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.185iT - 3T^{2} \)
7 \( 1 - 4.45iT - 7T^{2} \)
11 \( 1 - 2.64T + 11T^{2} \)
13 \( 1 - 1.30iT - 13T^{2} \)
17 \( 1 - 3.51iT - 17T^{2} \)
23 \( 1 + 6.52iT - 23T^{2} \)
29 \( 1 - 5.20T + 29T^{2} \)
31 \( 1 - 10.8T + 31T^{2} \)
37 \( 1 + 2.04iT - 37T^{2} \)
41 \( 1 + 3.80T + 41T^{2} \)
43 \( 1 - 4.77iT - 43T^{2} \)
47 \( 1 + 1.49iT - 47T^{2} \)
53 \( 1 - 0.225iT - 53T^{2} \)
59 \( 1 - 2.86T + 59T^{2} \)
61 \( 1 + 6.31T + 61T^{2} \)
67 \( 1 - 13.1iT - 67T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + 5.42iT - 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 - 3.84iT - 83T^{2} \)
89 \( 1 + 1.67T + 89T^{2} \)
97 \( 1 + 9.48iT - 97T^{2} \)
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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

−8.444453861376146926656748844189, −8.254976954285041557007936632237, −6.91809813671018492459607508043, −6.44551850288340189648181054241, −5.79660741490129842053923833808, −4.74396943909373835547124423574, −4.19396331986799826934397194362, −2.97963990119312507033193773447, −2.17512797071609391265995140089, −1.20402078652065491988634112902, 0.790436429241349998637711532800, 1.48046136886080260873656137154, 3.02402740951813655046822517130, 3.83339299454468061361415914394, 4.46222295004069956802293413275, 5.15652668551637110641534906989, 6.44650205928610849453902103731, 6.91226890537512336053500537865, 7.55727742498734139752095880571, 8.181892987502643598263382149842

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