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Properties

Label 2-1900-475.341-c0-0-0
Degree $2$
Conductor $1900$
Sign $0.604 - 0.796i$
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  + (0.913 + 0.406i)5-s − 0.209·7-s + (−0.809 + 0.587i)9-s + (0.169 + 0.122i)11-s + (−0.604 + 1.86i)17-s + (0.309 − 0.951i)19-s + (1.30 + 0.951i)23-s + (0.669 + 0.743i)25-s + (−0.190 − 0.0850i)35-s + 1.33·43-s + (−0.978 + 0.207i)45-s + (−0.309 − 0.951i)47-s − 0.956·49-s + (0.104 + 0.181i)55-s + (−1.08 − 0.786i)61-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)5-s − 0.209·7-s + (−0.809 + 0.587i)9-s + (0.169 + 0.122i)11-s + (−0.604 + 1.86i)17-s + (0.309 − 0.951i)19-s + (1.30 + 0.951i)23-s + (0.669 + 0.743i)25-s + (−0.190 − 0.0850i)35-s + 1.33·43-s + (−0.978 + 0.207i)45-s + (−0.309 − 0.951i)47-s − 0.956·49-s + (0.104 + 0.181i)55-s + (−1.08 − 0.786i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.604 - 0.796i$
Analytic conductor: \(0.948223\)
Root analytic conductor: \(0.973767\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :0),\ 0.604 - 0.796i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.230628257\)
\(L(\frac12)\) \(\approx\) \(1.230628257\)
\(L(1)\) 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 \( 1 + (-0.913 - 0.406i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
good3 \( 1 + (0.809 - 0.587i)T^{2} \)
7 \( 1 + 0.209T + T^{2} \)
11 \( 1 + (-0.169 - 0.122i)T + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.604 - 1.86i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 - 1.33T + T^{2} \)
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 \( 1 + (1.08 + 0.786i)T + (0.309 + 0.951i)T^{2} \)
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

−9.384942045391120625539208916536, −8.903480321813861435464527176956, −8.004182237893729903863062206207, −7.03416549840858707651559993398, −6.31126423671359152931352628834, −5.58929483002941659668875152144, −4.79848120215114808085245127554, −3.52322342757056530851500717599, −2.62120132765566967346987971891, −1.64496422860681170422904918114, 0.956568679633846329940462433716, 2.46892298113747544709107222301, 3.18233632467201119368798440317, 4.54510092321117087751966476915, 5.28840486892329587649307669664, 6.12986602211642625729575313661, 6.74251915974446348885405839760, 7.74321867160411728119097302640, 8.855853679953283466692252594252, 9.168620068137830440401217263893

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