Properties

Label 2-1900-19.18-c2-0-17
Degree $2$
Conductor $1900$
Sign $-0.887 - 0.461i$
Analytic cond. $51.7712$
Root an. cond. $7.19522$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.26i·3-s + 6.30·7-s − 1.67·9-s + 4.53·11-s + 11.5i·13-s − 1.08·17-s + (−16.8 − 8.76i)19-s + 20.5i·21-s − 31.5·23-s + 23.9i·27-s + 38.1i·29-s − 58.2i·31-s + 14.8i·33-s + 40.2i·37-s − 37.6·39-s + ⋯
L(s)  = 1  + 1.08i·3-s + 0.900·7-s − 0.186·9-s + 0.411·11-s + 0.887i·13-s − 0.0638·17-s + (−0.887 − 0.461i)19-s + 0.980i·21-s − 1.37·23-s + 0.886i·27-s + 1.31i·29-s − 1.87i·31-s + 0.448i·33-s + 1.08i·37-s − 0.966·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 - 0.461i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.887 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.887 - 0.461i$
Analytic conductor: \(51.7712\)
Root analytic conductor: \(7.19522\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1),\ -0.887 - 0.461i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.729338264\)
\(L(\frac12)\) \(\approx\) \(1.729338264\)
\(L(2)\) 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 \( 1 + (16.8 + 8.76i)T \)
good3 \( 1 - 3.26iT - 9T^{2} \)
7 \( 1 - 6.30T + 49T^{2} \)
11 \( 1 - 4.53T + 121T^{2} \)
13 \( 1 - 11.5iT - 169T^{2} \)
17 \( 1 + 1.08T + 289T^{2} \)
23 \( 1 + 31.5T + 529T^{2} \)
29 \( 1 - 38.1iT - 841T^{2} \)
31 \( 1 + 58.2iT - 961T^{2} \)
37 \( 1 - 40.2iT - 1.36e3T^{2} \)
41 \( 1 - 18.0iT - 1.68e3T^{2} \)
43 \( 1 + 2.74T + 1.84e3T^{2} \)
47 \( 1 - 76.1T + 2.20e3T^{2} \)
53 \( 1 - 8.06iT - 2.80e3T^{2} \)
59 \( 1 - 95.3iT - 3.48e3T^{2} \)
61 \( 1 + 4.28T + 3.72e3T^{2} \)
67 \( 1 - 70.7iT - 4.48e3T^{2} \)
71 \( 1 - 93.3iT - 5.04e3T^{2} \)
73 \( 1 + 14.2T + 5.32e3T^{2} \)
79 \( 1 + 6.11iT - 6.24e3T^{2} \)
83 \( 1 - 89.0T + 6.88e3T^{2} \)
89 \( 1 + 92.3iT - 7.92e3T^{2} \)
97 \( 1 - 64.6iT - 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.322758943280324637122341789086, −8.777759484540039237905362525429, −7.926260921809397181645631831952, −7.00198138161208505443960999278, −6.09214333107184675418252919440, −5.12264613602001905615886436185, −4.25118374382459368148639957114, −3.99646000389235232746774059255, −2.49701859030550024407408852040, −1.41790005864460272246218372347, 0.43650133033859319990376703056, 1.61549287917761063573478997258, 2.27699923536294836578271579918, 3.69105918730999876530182658645, 4.60031866690565513880670579804, 5.67235243909531580680856408434, 6.36157538038380662959137314341, 7.20365477278624377338969301776, 8.030568624643544302036979779258, 8.283333907757947669873537215865

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