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Properties

Label 2-1900-19.7-c1-0-14
Degree $2$
Conductor $1900$
Sign $0.998 - 0.0593i$
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  + (−0.176 − 0.306i)3-s + 4.30·7-s + (1.43 − 2.48i)9-s + 6.01·11-s + (−2.97 + 5.15i)13-s + (1.93 + 3.35i)17-s + (4.19 − 1.17i)19-s + (−0.760 − 1.31i)21-s + (0.391 − 0.678i)23-s − 2.07·27-s + (−3.98 + 6.89i)29-s − 4.49·31-s + (−1.06 − 1.84i)33-s + 0.988·37-s + 2.10·39-s + ⋯
L(s)  = 1  + (−0.102 − 0.176i)3-s + 1.62·7-s + (0.479 − 0.829i)9-s + 1.81·11-s + (−0.825 + 1.42i)13-s + (0.469 + 0.813i)17-s + (0.963 − 0.268i)19-s + (−0.166 − 0.287i)21-s + (0.0816 − 0.141i)23-s − 0.399·27-s + (−0.739 + 1.28i)29-s − 0.806·31-s + (−0.184 − 0.320i)33-s + 0.162·37-s + 0.336·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.998 - 0.0593i$
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: \((2,\ 1900,\ (\ :1/2),\ 0.998 - 0.0593i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.374451558\)
\(L(\frac12)\) \(\approx\) \(2.374451558\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.19 + 1.17i)T \)
good3 \( 1 + (0.176 + 0.306i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 4.30T + 7T^{2} \)
11 \( 1 - 6.01T + 11T^{2} \)
13 \( 1 + (2.97 - 5.15i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.93 - 3.35i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.391 + 0.678i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.98 - 6.89i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.49T + 31T^{2} \)
37 \( 1 - 0.988T + 37T^{2} \)
41 \( 1 + (3.15 + 5.46i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.785 + 1.35i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.630 - 1.09i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.07 - 7.05i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.62 + 4.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.80 - 4.85i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.52 + 6.09i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.90 - 5.03i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.62 + 8.00i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.99 - 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.58T + 83T^{2} \)
89 \( 1 + (-1.69 + 2.93i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.69 + 6.40i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.170814045389700872549909098751, −8.615665986997794545772718701224, −7.39294122448085751332443778932, −7.05195466584209128476320834926, −6.11485531674369764711261857491, −5.06942801496316408694755338277, −4.26650303086279748338769461326, −3.58226047882359449999838266827, −1.77110639019049965939970615945, −1.35277737591397920257655032417, 1.10266223427222463590687552903, 2.04135823160590681669107460058, 3.39446107394701473249464181313, 4.45467586822956731016574868253, 5.09772652737599972975825269469, 5.76196421480094750710800257255, 7.09750219006934941974534767655, 7.72501547675270635140438813438, 8.190249910359632009123816127010, 9.368784596541782353327339116880

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