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Properties

Label 2-1900-19.7-c1-0-22
Degree $2$
Conductor $1900$
Sign $-0.200 + 0.979i$
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  + (−0.182 − 0.315i)3-s − 0.635·7-s + (1.43 − 2.48i)9-s + 1.63·11-s + (−0.5 + 0.866i)13-s + (−3.29 − 5.71i)17-s + (0.0466 + 4.35i)19-s + (0.115 + 0.200i)21-s + (0.433 − 0.750i)23-s − 2.13·27-s + (4.54 − 7.87i)29-s − 1.86·31-s + (−0.298 − 0.516i)33-s − 0.635·37-s + 0.364·39-s + ⋯
L(s)  = 1  + (−0.105 − 0.182i)3-s − 0.240·7-s + (0.477 − 0.827i)9-s + 0.493·11-s + (−0.138 + 0.240i)13-s + (−0.799 − 1.38i)17-s + (0.0107 + 0.999i)19-s + (0.0252 + 0.0437i)21-s + (0.0904 − 0.156i)23-s − 0.411·27-s + (0.844 − 1.46i)29-s − 0.335·31-s + (−0.0518 − 0.0898i)33-s − 0.104·37-s + 0.0583·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.200 + 0.979i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ -0.200 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.305716615\)
\(L(\frac12)\) \(\approx\) \(1.305716615\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0466 - 4.35i)T \)
good3 \( 1 + (0.182 + 0.315i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 0.635T + 7T^{2} \)
11 \( 1 - 1.63T + 11T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.29 + 5.71i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.433 + 0.750i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.54 + 7.87i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.86T + 31T^{2} \)
37 \( 1 + 0.635T + 37T^{2} \)
41 \( 1 + (0.953 + 1.65i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.98 - 3.42i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.54 + 7.87i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.93 - 8.54i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.54 + 7.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.13 - 1.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.23 + 10.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.25 + 2.16i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.201 + 0.349i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.36 + 9.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.8T + 83T^{2} \)
89 \( 1 + (0.271 - 0.469i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.68 + 6.38i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.260048222894897462784403568338, −8.179506241286662885724038281699, −7.31463863895944585531254849135, −6.58391538074751781135919344144, −6.02430922677181674475015996309, −4.81013565446039335723577139172, −4.04800762536546594508392045534, −3.06304328301156417843871842410, −1.84977692368050400955987286579, −0.49817736071315285433076793432, 1.39558637589668035482821356872, 2.52960893044704450359445101485, 3.69259709147745909897473917887, 4.56455337388108990245348004508, 5.29708256674817340884022456210, 6.38208140879216496727015681130, 6.98068149217595453306604911597, 7.907194179703303685419692483391, 8.710591352738453411660210478252, 9.376667225979218761475236195329

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