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Properties

Label 2-1900-475.436-c0-0-0
Degree $2$
Conductor $1900$
Sign $0.387 - 0.921i$
Analytic cond. $0.948223$
Root an. cond. $0.973767$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.104 + 0.994i)5-s + 1.82·7-s + (−0.809 − 0.587i)9-s + (−1.47 + 1.07i)11-s + (0.413 + 1.27i)17-s + (0.309 + 0.951i)19-s + (1.30 − 0.951i)23-s + (−0.978 − 0.207i)25-s + (−0.190 + 1.81i)35-s − 1.95·43-s + (0.669 − 0.743i)45-s + (−0.309 + 0.951i)47-s + 2.33·49-s + (−0.913 − 1.58i)55-s + (1.58 − 1.14i)61-s + ⋯
L(s)  = 1  + (−0.104 + 0.994i)5-s + 1.82·7-s + (−0.809 − 0.587i)9-s + (−1.47 + 1.07i)11-s + (0.413 + 1.27i)17-s + (0.309 + 0.951i)19-s + (1.30 − 0.951i)23-s + (−0.978 − 0.207i)25-s + (−0.190 + 1.81i)35-s − 1.95·43-s + (0.669 − 0.743i)45-s + (−0.309 + 0.951i)47-s + 2.33·49-s + (−0.913 − 1.58i)55-s + (1.58 − 1.14i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.387 - 0.921i$
Analytic conductor: \(0.948223\)
Root analytic conductor: \(0.973767\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1861, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :0),\ 0.387 - 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.179463880\)
\(L(\frac12)\) \(\approx\) \(1.179463880\)
\(L(1)\) 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 \( 1 + (0.104 - 0.994i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (0.809 + 0.587i)T^{2} \)
7 \( 1 - 1.82T + T^{2} \)
11 \( 1 + (1.47 - 1.07i)T + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.413 - 1.27i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + 1.95T + T^{2} \)
47 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-1.58 + 1.14i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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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

−9.710009290362181276316828413603, −8.345248230680674185320636775281, −8.130168454911555029880143827207, −7.34164734963892603866965823743, −6.41541814247999582704642625758, −5.41245964774032406956058614312, −4.81865561328176641689260153488, −3.67283033647532118699740048407, −2.62866972466538092760993707399, −1.71228496827695187931748051540, 0.912382448147095896730278444607, 2.25974538650321310691659273504, 3.24634210267969985332637367493, 4.81988173236302244613455357025, 5.19352408786289804953531505884, 5.49236733893881082552420806842, 7.23668846902493335347689624635, 7.86973935496987571771981849942, 8.475167097148167537148945496359, 8.899554874781983254248300279775

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