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Properties

Label 2-1900-19.7-c1-0-23
Degree $2$
Conductor $1900$
Sign $0.440 + 0.897i$
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  + (−0.226 − 0.392i)3-s + 2.54·7-s + (1.39 − 2.42i)9-s + 2.22·11-s + (3.51 − 6.08i)13-s + (1.27 + 2.21i)17-s + (−2.70 − 3.41i)19-s + (−0.575 − 0.997i)21-s + (−4.01 + 6.95i)23-s − 2.62·27-s + (0.941 − 1.63i)29-s + 5.98·31-s + (−0.503 − 0.872i)33-s − 2.86·37-s − 3.17·39-s + ⋯
L(s)  = 1  + (−0.130 − 0.226i)3-s + 0.961·7-s + (0.465 − 0.806i)9-s + 0.670·11-s + (0.973 − 1.68i)13-s + (0.310 + 0.537i)17-s + (−0.620 − 0.784i)19-s + (−0.125 − 0.217i)21-s + (−0.837 + 1.44i)23-s − 0.505·27-s + (0.174 − 0.302i)29-s + 1.07·31-s + (−0.0876 − 0.151i)33-s − 0.470·37-s − 0.509·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.440 + 0.897i$
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: \((2,\ 1900,\ (\ :1/2),\ 0.440 + 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.041921916\)
\(L(\frac12)\) \(\approx\) \(2.041921916\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.70 + 3.41i)T \)
good3 \( 1 + (0.226 + 0.392i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 2.54T + 7T^{2} \)
11 \( 1 - 2.22T + 11T^{2} \)
13 \( 1 + (-3.51 + 6.08i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.27 - 2.21i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (4.01 - 6.95i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.941 + 1.63i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.98T + 31T^{2} \)
37 \( 1 + 2.86T + 37T^{2} \)
41 \( 1 + (3.67 + 6.36i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.84 + 3.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.36 - 4.09i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.14 - 8.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.73 - 6.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.17 + 7.23i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.17 + 10.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.13 + 7.16i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.32 - 10.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.13 - 3.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.7T + 83T^{2} \)
89 \( 1 + (-7.19 + 12.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.91 - 5.04i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.979908596535269049960722028139, −8.193456196204335618304094560373, −7.67548773134447867898408322484, −6.59237223993798687917538960186, −5.95463024400532498775611507088, −5.09403313293487293061482384511, −4.00299662258085121297155045586, −3.30797942139203424275878683151, −1.78472206673415593598038184121, −0.854166870613623228559787421149, 1.42456140929434165814599837784, 2.16337488967381672117901750139, 3.78991619708252612792241069721, 4.45617688748083034123012595384, 5.09409729381329876624722848424, 6.39051525825552787876396257025, 6.76697586857695187367473735624, 8.133278227921016298696138682538, 8.324546006386704404916160599536, 9.366297589931398000332583341097

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