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Properties

Label 2-1900-95.49-c1-0-11
Degree $2$
Conductor $1900$
Sign $-0.498 - 0.867i$
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  + (2.77 + 1.60i)3-s + 2.20i·7-s + (3.63 + 6.29i)9-s − 1.20·11-s + (−0.866 + 0.5i)13-s + (−1.85 − 1.07i)17-s + (−4.30 − 0.673i)19-s + (−3.53 + 6.11i)21-s + (−8.02 + 4.63i)23-s + 13.6i·27-s + (4.16 + 7.21i)29-s + 8.26·31-s + (−3.34 − 1.92i)33-s + 2.20i·37-s − 3.20·39-s + ⋯
L(s)  = 1  + (1.60 + 0.925i)3-s + 0.833i·7-s + (1.21 + 2.09i)9-s − 0.363·11-s + (−0.240 + 0.138i)13-s + (−0.449 − 0.259i)17-s + (−0.988 − 0.154i)19-s + (−0.770 + 1.33i)21-s + (−1.67 + 0.966i)23-s + 2.63i·27-s + (0.773 + 1.33i)29-s + 1.48·31-s + (−0.581 − 0.335i)33-s + 0.362i·37-s − 0.513·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.498 - 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.498 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.498 - 0.867i$
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: \((2,\ 1900,\ (\ :1/2),\ -0.498 - 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.692386530\)
\(L(\frac12)\) \(\approx\) \(2.692386530\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.30 + 0.673i)T \)
good3 \( 1 + (-2.77 - 1.60i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 - 2.20iT - 7T^{2} \)
11 \( 1 + 1.20T + 11T^{2} \)
13 \( 1 + (0.866 - 0.5i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.85 + 1.07i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (8.02 - 4.63i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.16 - 7.21i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.26T + 31T^{2} \)
37 \( 1 - 2.20iT - 37T^{2} \)
41 \( 1 + (-3.30 + 5.72i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.03 + 1.17i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.21 + 4.16i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.232 + 0.134i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.16 - 7.21i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.70 - 2.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.84 + 1.06i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.23 + 9.06i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.20 - 2.42i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.20 + 14.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.73iT - 83T^{2} \)
89 \( 1 + (5.40 + 9.36i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.6 - 7.87i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.169940707133551118818303110173, −8.874482334755279521423186540867, −8.138576136781481362475073176785, −7.45818637068316299243435779503, −6.30060068604264536403664687525, −5.17433477450695156431219263839, −4.42350200874000904473138253266, −3.57583109611646783578014420473, −2.60780804010186684673832290669, −2.03091643410842390662532621536, 0.75848869116508844107064401442, 2.13458591754623908246234831743, 2.67852538842759472828819459813, 3.93952526303508092255428064917, 4.41499819923326684665332420642, 6.21051985465142360144767914331, 6.65751232934232647756798721867, 7.67871824797537442149239418808, 8.133828515807376099766920611091, 8.603141279854200837316573093790

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