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Properties

Label 2-1900-19.11-c1-0-21
Degree $2$
Conductor $1900$
Sign $0.345 + 0.938i$
Analytic cond. $15.1715$
Root an. cond. $3.89507$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.789 + 1.36i)3-s + 1.39·7-s + (0.254 + 0.440i)9-s − 4.67·11-s + (−0.696 − 1.20i)13-s + (2.17 − 3.76i)17-s + (−0.608 − 4.31i)19-s + (−1.10 + 1.91i)21-s + (−1.64 − 2.84i)23-s − 5.53·27-s + (−1.19 − 2.07i)29-s − 3.94·31-s + (3.69 − 6.39i)33-s + 11.3·37-s + 2.19·39-s + ⋯
L(s)  = 1  + (−0.455 + 0.789i)3-s + 0.529·7-s + (0.0848 + 0.146i)9-s − 1.41·11-s + (−0.193 − 0.334i)13-s + (0.527 − 0.913i)17-s + (−0.139 − 0.990i)19-s + (−0.241 + 0.417i)21-s + (−0.342 − 0.592i)23-s − 1.06·27-s + (−0.222 − 0.384i)29-s − 0.707·31-s + (0.642 − 1.11i)33-s + 1.86·37-s + 0.352·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.345 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.345 + 0.938i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ 0.345 + 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8202380174\)
\(L(\frac12)\) \(\approx\) \(0.8202380174\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 + (0.608 + 4.31i)T \)
good3 \( 1 + (0.789 - 1.36i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 1.39T + 7T^{2} \)
11 \( 1 + 4.67T + 11T^{2} \)
13 \( 1 + (0.696 + 1.20i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.17 + 3.76i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.64 + 2.84i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.19 + 2.07i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.94T + 31T^{2} \)
37 \( 1 - 11.3T + 37T^{2} \)
41 \( 1 + (5.99 - 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.68 + 6.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.48 + 2.56i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.81 + 6.60i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.34 + 9.25i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.146 - 0.254i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.77 + 6.53i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.0920 - 0.159i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.919 + 1.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.875 - 1.51i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.49T + 83T^{2} \)
89 \( 1 + (-1.02 - 1.77i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.261 - 0.453i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.286262498328942407800340884146, −8.026112914413699701079932365913, −7.77737359545722415947911159334, −6.66021834457584092583165563435, −5.49595459098967467102876527245, −5.02366628559455559593384512619, −4.40000837490301255940093438606, −3.10494808948247593941125037380, −2.15579976551802782588943817107, −0.32562824785944790071643741287, 1.28013840739616645834945662425, 2.19421559132422105723708242193, 3.51094538461667496647346132047, 4.52480085077498084411593192173, 5.66204988335987431132550278529, 5.97709908505690393731043010554, 7.17771831705053670470867677232, 7.72149607345464085012134386593, 8.302372415733003563674559240680, 9.417702715427788755433564505115

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