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Properties

Label 2-1900-475.94-c0-0-1
Degree $2$
Conductor $1900$
Sign $-0.985 + 0.166i$
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  + (−0.978 + 0.207i)5-s − 1.48i·7-s + (−0.309 + 0.951i)9-s + (−0.413 − 1.27i)11-s + (−1.16 + 1.60i)17-s + (−0.809 − 0.587i)19-s + (−1.80 + 0.587i)23-s + (0.913 − 0.406i)25-s + (0.309 + 1.45i)35-s + 0.813i·43-s + (0.104 − 0.994i)45-s + (−1.01 − 1.40i)47-s − 1.20·49-s + (0.669 + 1.15i)55-s + (−0.564 − 1.73i)61-s + ⋯
L(s)  = 1  + (−0.978 + 0.207i)5-s − 1.48i·7-s + (−0.309 + 0.951i)9-s + (−0.413 − 1.27i)11-s + (−1.16 + 1.60i)17-s + (−0.809 − 0.587i)19-s + (−1.80 + 0.587i)23-s + (0.913 − 0.406i)25-s + (0.309 + 1.45i)35-s + 0.813i·43-s + (0.104 − 0.994i)45-s + (−1.01 − 1.40i)47-s − 1.20·49-s + (0.669 + 1.15i)55-s + (−0.564 − 1.73i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.985 + 0.166i$
Analytic conductor: \(0.948223\)
Root analytic conductor: \(0.973767\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :0),\ -0.985 + 0.166i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1798034129\)
\(L(\frac12)\) \(\approx\) \(0.1798034129\)
\(L(1)\) 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 \( 1 + (0.978 - 0.207i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (0.309 - 0.951i)T^{2} \)
7 \( 1 + 1.48iT - T^{2} \)
11 \( 1 + (0.413 + 1.27i)T + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (1.16 - 1.60i)T + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - 0.813iT - T^{2} \)
47 \( 1 + (1.01 + 1.40i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.564 + 1.73i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.309 + 0.951i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (1.64 - 0.535i)T + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.690 + 0.951i)T + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.309 - 0.951i)T^{2} \)
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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

−8.675650874427007688175957033104, −8.160866347648314668374270755347, −7.68349873989889546433647670346, −6.69331745547453113824572633262, −5.98520389068630497159537837294, −4.69668058518647895864296572001, −4.04113542258265255516236583976, −3.31525132608871995910993042484, −1.93573783561039870434267476182, −0.12186651911457227438829067125, 2.09673600407652630882622861992, 2.89602684164907475972188895170, 4.16575879498153142457227689472, 4.77591984841608821822988268847, 5.82700181389947837654529197326, 6.60934976540499657050949272142, 7.47827387466602745628058158341, 8.330057005632079491448166618810, 8.962718149903239790678329411862, 9.542935888336655934085259010042

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