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Properties

Label 2-1900-19.11-c1-0-8
Degree $2$
Conductor $1900$
Sign $0.929 - 0.370i$
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  + (0.932 − 1.61i)3-s − 3.93·7-s + (−0.240 − 0.416i)9-s − 2.01·11-s + (3.09 + 5.36i)13-s + (2.28 − 3.95i)17-s + (−4.29 + 0.721i)19-s + (−3.66 + 6.34i)21-s + (4.32 + 7.48i)23-s + 4.70·27-s + (2.59 + 4.49i)29-s − 0.856·31-s + (−1.88 + 3.25i)33-s + 6.03·37-s + 11.5·39-s + ⋯
L(s)  = 1  + (0.538 − 0.932i)3-s − 1.48·7-s + (−0.0800 − 0.138i)9-s − 0.608·11-s + (0.859 + 1.48i)13-s + (0.553 − 0.958i)17-s + (−0.986 + 0.165i)19-s + (−0.800 + 1.38i)21-s + (0.901 + 1.56i)23-s + 0.904·27-s + (0.482 + 0.835i)29-s − 0.153·31-s + (−0.327 + 0.567i)33-s + 0.991·37-s + 1.85·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.929 - 0.370i$
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: \((2,\ 1900,\ (\ :1/2),\ 0.929 - 0.370i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.573333144\)
\(L(\frac12)\) \(\approx\) \(1.573333144\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.29 - 0.721i)T \)
good3 \( 1 + (-0.932 + 1.61i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 3.93T + 7T^{2} \)
11 \( 1 + 2.01T + 11T^{2} \)
13 \( 1 + (-3.09 - 5.36i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.28 + 3.95i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-4.32 - 7.48i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.59 - 4.49i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.856T + 31T^{2} \)
37 \( 1 - 6.03T + 37T^{2} \)
41 \( 1 + (1.37 - 2.37i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.39 + 11.0i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.32 + 7.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.17 - 5.49i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.18 - 12.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.73 - 8.20i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.24 - 9.08i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.51 - 4.35i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.663 - 1.14i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.38 + 2.39i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.5T + 83T^{2} \)
89 \( 1 + (3.45 + 5.98i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.02 - 3.50i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.064371225975437901432186324353, −8.635205886086350866514574828473, −7.40811601511485262420324548718, −7.06438286287693267356202241175, −6.31661356442468868394312230624, −5.40050157845689081291230904177, −4.12280977163134063907047351436, −3.18069953937846446749671266841, −2.36370276496215595261655270891, −1.16534473840707191626123107922, 0.60242435965642954920382380141, 2.67913602072430887976786035052, 3.24302381231917822960661051240, 4.02238731706401748713157909130, 4.97942802687377816561861213540, 6.24589111250835245715568020774, 6.40805504665382095845974714706, 7.973263891779836756197046094296, 8.354024942325685421489300704906, 9.364543078076935006231043992447

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