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Properties

Label 2-1900-95.64-c1-0-23
Degree $2$
Conductor $1900$
Sign $0.640 + 0.767i$
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  + (2.74 − 1.58i)3-s + 1.53i·7-s + (3.50 − 6.07i)9-s − 3.79·11-s + (5.80 + 3.34i)13-s + (5.21 − 3.00i)17-s + (3.93 − 1.87i)19-s + (2.42 + 4.20i)21-s + (−3.79 − 2.19i)23-s − 12.7i·27-s + (−0.549 + 0.951i)29-s − 1.05·31-s + (−10.4 + 6.00i)33-s + 10.7i·37-s + 21.2·39-s + ⋯
L(s)  = 1  + (1.58 − 0.913i)3-s + 0.579i·7-s + (1.16 − 2.02i)9-s − 1.14·11-s + (1.60 + 0.928i)13-s + (1.26 − 0.729i)17-s + (0.902 − 0.430i)19-s + (0.529 + 0.917i)21-s + (−0.791 − 0.457i)23-s − 2.44i·27-s + (−0.102 + 0.176i)29-s − 0.189·31-s + (−1.81 + 1.04i)33-s + 1.77i·37-s + 3.39·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.640 + 0.767i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ 0.640 + 0.767i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.264659056\)
\(L(\frac12)\) \(\approx\) \(3.264659056\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.93 + 1.87i)T \)
good3 \( 1 + (-2.74 + 1.58i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 - 1.53iT - 7T^{2} \)
11 \( 1 + 3.79T + 11T^{2} \)
13 \( 1 + (-5.80 - 3.34i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-5.21 + 3.00i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.79 + 2.19i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.549 - 0.951i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.05T + 31T^{2} \)
37 \( 1 - 10.7iT - 37T^{2} \)
41 \( 1 + (1.76 + 3.05i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.80 + 2.77i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (9.73 + 5.61i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.36 + 0.788i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.61 - 2.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.0331 - 0.0574i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.94 + 2.85i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.23 - 5.60i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-12.1 + 6.98i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.996 + 1.72i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.3iT - 83T^{2} \)
89 \( 1 + (-1.53 + 2.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.68 - 4.43i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.915863146935191669504145255773, −8.283485506290429033222173836544, −7.74814489867646480000028065876, −6.94208064702522864738224360030, −6.11413203188529973187833783085, −5.07371169458476901151469513125, −3.70419656303561619406083435916, −3.05772450063551588369520005288, −2.18406479684249371499446573633, −1.16765070659586012016622681824, 1.40585049020022463081376557475, 2.73287822909256848874783519886, 3.57535708953068813737694857810, 3.92348699494138031987568888339, 5.22523519357008488293452040383, 5.91060290600904364191707961095, 7.50536050311709730565317720291, 7.971153138224708653351433222070, 8.340857857648889121134200060240, 9.422052391660893406638449927850

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