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Properties

Label 2-3800-5.4-c1-0-26
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  + 0.347i·3-s + 3.41i·7-s + 2.87·9-s + 2.41·11-s + 6.29i·13-s + 2.34i·17-s − 19-s − 1.18·21-s + 2.49i·23-s + 2.04i·27-s + 8.17·29-s − 2.77·31-s + 0.837i·33-s + 0.977i·37-s − 2.18·39-s + ⋯
L(s)  = 1  + 0.200i·3-s + 1.28i·7-s + 0.959·9-s + 0.727·11-s + 1.74i·13-s + 0.569i·17-s − 0.229·19-s − 0.258·21-s + 0.519i·23-s + 0.392i·27-s + 1.51·29-s − 0.498·31-s + 0.145i·33-s + 0.160i·37-s − 0.349·39-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

\(L(1)\) \(\approx\) \(1.989329724\)
\(L(\frac12)\) \(\approx\) \(1.989329724\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.347iT - 3T^{2} \)
7 \( 1 - 3.41iT - 7T^{2} \)
11 \( 1 - 2.41T + 11T^{2} \)
13 \( 1 - 6.29iT - 13T^{2} \)
17 \( 1 - 2.34iT - 17T^{2} \)
23 \( 1 - 2.49iT - 23T^{2} \)
29 \( 1 - 8.17T + 29T^{2} \)
31 \( 1 + 2.77T + 31T^{2} \)
37 \( 1 - 0.977iT - 37T^{2} \)
41 \( 1 + 3.49T + 41T^{2} \)
43 \( 1 + 2.75iT - 43T^{2} \)
47 \( 1 + 6.29iT - 47T^{2} \)
53 \( 1 - 2.38iT - 53T^{2} \)
59 \( 1 + 3.67T + 59T^{2} \)
61 \( 1 + 12.7T + 61T^{2} \)
67 \( 1 + 2.41iT - 67T^{2} \)
71 \( 1 - 4.51T + 71T^{2} \)
73 \( 1 - 1.81iT - 73T^{2} \)
79 \( 1 - 5.04T + 79T^{2} \)
83 \( 1 + 8.07iT - 83T^{2} \)
89 \( 1 - 2.94T + 89T^{2} \)
97 \( 1 - 3.09iT - 97T^{2} \)
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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

−8.942763432709325135908630866790, −8.154230513918400660110746195226, −7.07356802786268792572702282622, −6.55993858303700047524542450020, −5.85505911803266729765354787691, −4.82532741913189598420109175174, −4.24516276357880618264776846238, −3.36571280805303564520261683484, −2.12171625357295156442403259898, −1.50601740205259182684811960351, 0.62440370078852320082160612569, 1.35528109140582765790626247289, 2.76552139917970842833926793925, 3.65071992631756074727208803121, 4.42115863555398423969183010761, 5.10173832176468083521331896749, 6.27863755806285691844212619052, 6.79034462353144567564949155096, 7.62005158822670028803309901470, 7.975727670159443995970786172481

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