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Properties

Label 2-1900-19.7-c1-0-10
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

\(L(1)\) \(\approx\) \(2.351302961\)
\(L(\frac12)\) \(\approx\) \(2.351302961\)
\(L(\frac{3}{2})\) 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

−9.357257572175050846135927198901, −8.344341184861610423942153765999, −8.075427679038430607476533032213, −7.19593914263398181071168189059, −6.01869979583111563298584421443, −5.05390515212952000677141934368, −4.51563182560882197081618859974, −3.64467364868950506884481280897, −2.37848496650459809085554764447, −1.37991742072317234844212957327, 0.902198720099450803939195994870, 2.18585251577683595228738070661, 2.68763166669049564316522553378, 4.40589825296830309815275947923, 4.97481737193975414423511609423, 5.68443564783372776490628280307, 7.12669392166271640130808821790, 7.57138557740548667228498751971, 8.220627759056775115917251228334, 8.681290585335138277389594953469

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