Properties

Label 2-1900-95.49-c1-0-4
Degree $2$
Conductor $1900$
Sign $-0.157 - 0.987i$
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  + (−0.866 − 0.5i)3-s + (−1 − 1.73i)9-s − 4·11-s + (0.866 − 0.5i)13-s + (−2.59 − 1.5i)17-s + (4 − 1.73i)19-s + (−4.33 + 2.5i)23-s + 5i·27-s + (3.5 + 6.06i)29-s + 4·31-s + (3.46 + 2i)33-s + 10i·37-s − 0.999·39-s + (2.5 − 4.33i)41-s + (−4.33 − 2.5i)43-s + ⋯
L(s)  = 1  + (−0.499 − 0.288i)3-s + (−0.333 − 0.577i)9-s − 1.20·11-s + (0.240 − 0.138i)13-s + (−0.630 − 0.363i)17-s + (0.917 − 0.397i)19-s + (−0.902 + 0.521i)23-s + 0.962i·27-s + (0.649 + 1.12i)29-s + 0.718·31-s + (0.603 + 0.348i)33-s + 1.64i·37-s − 0.160·39-s + (0.390 − 0.676i)41-s + (−0.660 − 0.381i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.157 - 0.987i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ -0.157 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4997482847\)
\(L(\frac12)\) \(\approx\) \(0.4997482847\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4 + 1.73i)T \)
good3 \( 1 + (0.866 + 0.5i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.59 + 1.5i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (4.33 - 2.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.5 - 6.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 + (-2.5 + 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.33 + 2.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.06 - 3.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (9.52 - 5.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.59 - 1.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.5 - 9.52i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-12.9 - 7.5i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.5 - 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (-1.5 - 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.33 - 2.5i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.474845899092081373007359926483, −8.573410681573364443135521935595, −7.86496083870508855545834138839, −6.97207714571569297030231619815, −6.28669796569647541201119120229, −5.39563746090040951447782600225, −4.77992128051291513828435032891, −3.43902332191729874695336194360, −2.63789883043029607334265364147, −1.15314612026434374664625946431, 0.20949064793891694032468013396, 2.01622462587638313738674687362, 2.95528333170578977248903306998, 4.23072993316367225933470201142, 4.93835238718973273443240482952, 5.79643385685210807829096836275, 6.38356424081060353592215435370, 7.69462645057270352237091106508, 8.040467600251818577238386424764, 8.977774250639823198778620220186

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