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Properties

Label 2-1900-475.94-c0-0-0
Degree $2$
Conductor $1900$
Sign $0.348 - 0.937i$
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  + (0.669 + 0.743i)5-s − 0.415i·7-s + (−0.309 + 0.951i)9-s + (0.604 + 1.86i)11-s + (0.478 − 0.658i)17-s + (−0.809 − 0.587i)19-s + (−1.80 + 0.587i)23-s + (−0.104 + 0.994i)25-s + (0.309 − 0.278i)35-s − 1.98i·43-s + (−0.913 + 0.406i)45-s + (1.01 + 1.40i)47-s + 0.827·49-s + (−0.978 + 1.69i)55-s + (0.0646 + 0.198i)61-s + ⋯
L(s)  = 1  + (0.669 + 0.743i)5-s − 0.415i·7-s + (−0.309 + 0.951i)9-s + (0.604 + 1.86i)11-s + (0.478 − 0.658i)17-s + (−0.809 − 0.587i)19-s + (−1.80 + 0.587i)23-s + (−0.104 + 0.994i)25-s + (0.309 − 0.278i)35-s − 1.98i·43-s + (−0.913 + 0.406i)45-s + (1.01 + 1.40i)47-s + 0.827·49-s + (−0.978 + 1.69i)55-s + (0.0646 + 0.198i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.348 - 0.937i$
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: \((2,\ 1900,\ (\ :0),\ 0.348 - 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.222497175\)
\(L(\frac12)\) \(\approx\) \(1.222497175\)
\(L(1)\) 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 \( 1 + (-0.669 - 0.743i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (0.309 - 0.951i)T^{2} \)
7 \( 1 + 0.415iT - T^{2} \)
11 \( 1 + (-0.604 - 1.86i)T + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (-0.478 + 0.658i)T + (-0.309 - 0.951i)T^{2} \)
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 \( 1 + 1.98iT - T^{2} \)
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 \( 1 + (-0.0646 - 0.198i)T + (-0.809 + 0.587i)T^{2} \)
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

−9.700053406612323616039431337582, −8.935264790893169867798212275150, −7.67096838615528921328810218012, −7.31155546734760920374933967051, −6.47221339658136215995045182133, −5.56814183698814064287478199359, −4.66277676784394414481582170801, −3.81579190326311926648953685531, −2.42024620650199755723844400601, −1.87755114979876742397291770074, 0.923002701889769194106646355136, 2.20777749626878687877516319432, 3.47851685021764618595228966873, 4.16591826213188132074901213957, 5.54220118905011317257235001250, 6.06536813213546621541166800894, 6.43809268867663124724898637502, 8.138982886000086542635589267876, 8.456979235899726044180724126438, 9.140348866988574599238458725668

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