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Properties

Label 2-1900-19.7-c1-0-16
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: $\chi_{1900} (501, \cdot )$
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

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

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