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Properties

Label 2-1900-19.7-c1-0-0
Degree $2$
Conductor $1900$
Sign $0.481 - 0.876i$
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  + (−0.628 − 1.08i)3-s − 4.97·7-s + (0.710 − 1.23i)9-s − 3.85·11-s + (1.33 − 2.31i)13-s + (−1.29 − 2.24i)17-s + (−1.24 + 4.17i)19-s + (3.12 + 5.40i)21-s + (1.08 − 1.88i)23-s − 5.55·27-s + (−1.29 + 2.24i)29-s + 7.76·31-s + (2.42 + 4.19i)33-s + 2.75·37-s − 3.35·39-s + ⋯
L(s)  = 1  + (−0.362 − 0.628i)3-s − 1.87·7-s + (0.236 − 0.410i)9-s − 1.16·11-s + (0.370 − 0.641i)13-s + (−0.314 − 0.544i)17-s + (−0.285 + 0.958i)19-s + (0.681 + 1.18i)21-s + (0.226 − 0.392i)23-s − 1.06·27-s + (−0.240 + 0.416i)29-s + 1.39·31-s + (0.421 + 0.730i)33-s + 0.453·37-s − 0.537·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.481 - 0.876i$
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: \((2,\ 1900,\ (\ :1/2),\ 0.481 - 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4565903230\)
\(L(\frac12)\) \(\approx\) \(0.4565903230\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.24 - 4.17i)T \)
good3 \( 1 + (0.628 + 1.08i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 4.97T + 7T^{2} \)
11 \( 1 + 3.85T + 11T^{2} \)
13 \( 1 + (-1.33 + 2.31i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.29 + 2.24i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.08 + 1.88i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.29 - 2.24i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.76T + 31T^{2} \)
37 \( 1 - 2.75T + 37T^{2} \)
41 \( 1 + (-3.66 - 6.34i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.895 + 1.55i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.854 - 1.47i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.98 - 6.89i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.127 - 0.220i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.66 - 2.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.60 - 11.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.85 - 6.68i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.25 - 3.90i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.52 + 9.57i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.04T + 83T^{2} \)
89 \( 1 + (4.76 - 8.24i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.61 + 9.72i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.505582614516378864636400998848, −8.528351120234239173132616514675, −7.62105553032877430113555090472, −6.89351496100322685244123867618, −6.15993680721551845300556369430, −5.71040191199284747848927892288, −4.37926201760913613177655683642, −3.27336766886915249386738871824, −2.62069032237765450501868842299, −0.915402674481901085167559164739, 0.21844597315166422374873537165, 2.26047308967202102932056970719, 3.20773220784047122587945255167, 4.14748810820569513394798976453, 4.97301207571903969361801723591, 5.95534082210026964596776632532, 6.56644601522575739667810625938, 7.41295339963811614729423640152, 8.391894033199413670815567021857, 9.385034506465941304780458299069

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