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Properties

Label 2-1900-19.11-c1-0-19
Degree $2$
Conductor $1900$
Sign $0.717 + 0.696i$
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  + (0.923 − 1.59i)3-s + 1.72·7-s + (−0.204 − 0.354i)9-s + 2.66·11-s + (−0.544 − 0.943i)13-s + (−0.660 + 1.14i)17-s + (2.41 + 3.62i)19-s + (1.59 − 2.76i)21-s + (−0.704 − 1.21i)23-s + 4.78·27-s + (0.0446 + 0.0773i)29-s + 8.90·31-s + (2.45 − 4.25i)33-s + 3.32·37-s − 2.01·39-s + ⋯
L(s)  = 1  + (0.533 − 0.923i)3-s + 0.653·7-s + (−0.0682 − 0.118i)9-s + 0.803·11-s + (−0.151 − 0.261i)13-s + (−0.160 + 0.277i)17-s + (0.554 + 0.832i)19-s + (0.348 − 0.603i)21-s + (−0.146 − 0.254i)23-s + 0.920·27-s + (0.00829 + 0.0143i)29-s + 1.59·31-s + (0.428 − 0.741i)33-s + 0.547·37-s − 0.322·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.717 + 0.696i$
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: \((2,\ 1900,\ (\ :1/2),\ 0.717 + 0.696i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.508141576\)
\(L(\frac12)\) \(\approx\) \(2.508141576\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.41 - 3.62i)T \)
good3 \( 1 + (-0.923 + 1.59i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 1.72T + 7T^{2} \)
11 \( 1 - 2.66T + 11T^{2} \)
13 \( 1 + (0.544 + 0.943i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.660 - 1.14i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.704 + 1.21i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.0446 - 0.0773i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.90T + 31T^{2} \)
37 \( 1 - 3.32T + 37T^{2} \)
41 \( 1 + (0.296 - 0.514i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.31 + 2.27i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.51 + 6.08i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.275 - 0.477i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.07 - 7.05i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.207 + 0.359i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.11 + 3.65i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.36 + 2.36i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.53 + 7.84i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.86 - 10.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.76T + 83T^{2} \)
89 \( 1 + (0.132 + 0.229i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.843 - 1.46i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.899948333968116928130889336172, −8.139828792766555499897957342774, −7.74782530647527938071520749383, −6.81869041693902797494523426011, −6.13827484969234610197975589012, −5.03671562023508479872266997505, −4.13916764362518198816474986804, −3.00807288499840061083992024838, −1.96074520178096000600771186959, −1.11844917775460905460078765419, 1.17855288151161613670438961628, 2.58523639293903226815733397841, 3.50758590439981507954327061159, 4.47809883731735056902257286999, 4.88092387277186059790875764006, 6.16564620521040662907979079445, 6.95358043859077070802156044866, 7.930056085591980067617280793787, 8.657250425706590957852908143787, 9.445281011112643513712991000604

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