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Properties

Label 2-1900-475.246-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 − 1.95·7-s + (0.309 + 0.951i)9-s + (−0.604 + 1.86i)11-s + (−1.47 + 1.07i)17-s + (−0.809 + 0.587i)19-s + (0.190 − 0.587i)23-s + (−0.104 − 0.994i)25-s + (−1.30 + 1.45i)35-s − 0.209·43-s + (0.913 + 0.406i)45-s + (0.809 + 0.587i)47-s + 2.82·49-s + (0.978 + 1.69i)55-s + (−0.0646 + 0.198i)61-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)5-s − 1.95·7-s + (0.309 + 0.951i)9-s + (−0.604 + 1.86i)11-s + (−1.47 + 1.07i)17-s + (−0.809 + 0.587i)19-s + (0.190 − 0.587i)23-s + (−0.104 − 0.994i)25-s + (−1.30 + 1.45i)35-s − 0.209·43-s + (0.913 + 0.406i)45-s + (0.809 + 0.587i)47-s + 2.82·49-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: $\chi_{1900} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :0),\ -0.348 - 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6157253839\)
\(L(\frac12)\) \(\approx\) \(0.6157253839\)
\(L(1)\) 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 \( 1 + 1.95T + T^{2} \)
11 \( 1 + (0.604 - 1.86i)T + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (1.47 - 1.07i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 + 0.209T + T^{2} \)
47 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
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

−9.684804790140483080793694469653, −9.018690215250002404408444936851, −8.173953164508901407961936142432, −7.11018630610474474147004168876, −6.51101058406748575614743852181, −5.73235625159288982913902835417, −4.64082405902181401531915526424, −4.10342165060886491190830641064, −2.53543804531689072451829621519, −1.93194513664864831252879003522, 0.40962753579880184930645718525, 2.56019610223348062616543685430, 3.11594554935862544148515280692, 3.88214336155628197450283314049, 5.39926521858662689300789627022, 6.28340224627399444358030428912, 6.54475411905185056114624808950, 7.27958803898752901507456862451, 8.752937483824783633056489339447, 9.190887020284336321273738549700

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