↑

Properties

Label 2-1900-19.7-c1-0-13
Degree $2$
Conductor $1900$
Sign $-0.219 - 0.975i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.58 + 2.74i)3-s + 3.03·7-s + (−3.51 + 6.08i)9-s + 2.71·11-s + (1.95 − 3.38i)13-s + (−0.944 − 1.63i)17-s + (1.83 + 3.95i)19-s + (4.80 + 8.32i)21-s + (−3.57 + 6.19i)23-s − 12.7·27-s + (1.45 − 2.52i)29-s + 7.40·31-s + (4.30 + 7.45i)33-s − 3.43·37-s + 12.3·39-s + ⋯
L(s)  = 1  + (0.914 + 1.58i)3-s + 1.14·7-s + (−1.17 + 2.02i)9-s + 0.819·11-s + (0.542 − 0.939i)13-s + (−0.229 − 0.396i)17-s + (0.420 + 0.907i)19-s + (1.04 + 1.81i)21-s + (−0.745 + 1.29i)23-s − 2.45·27-s + (0.270 − 0.468i)29-s + 1.33·31-s + (0.748 + 1.29i)33-s − 0.564·37-s + 1.98·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.219 - 0.975i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ -0.219 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.875804233\)
\(L(\frac12)\) \(\approx\) \(2.875804233\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.83 - 3.95i)T \)
good3 \( 1 + (-1.58 - 2.74i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 3.03T + 7T^{2} \)
11 \( 1 - 2.71T + 11T^{2} \)
13 \( 1 + (-1.95 + 3.38i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.944 + 1.63i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (3.57 - 6.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.45 + 2.52i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.40T + 31T^{2} \)
37 \( 1 + 3.43T + 37T^{2} \)
41 \( 1 + (5.16 + 8.93i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.37 - 9.31i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.63 + 4.56i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.843 + 1.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.347 - 0.602i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.28 - 10.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.365 + 0.633i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.75 - 3.03i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.55 + 9.61i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.03 - 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 + (-4.34 + 7.51i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.43 + 5.94i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.479344205818833017716837191779, −8.623468359736121563680247981217, −8.176917022591008072046151851876, −7.45241443542654101189368011998, −5.91816590679615884257809979187, −5.22780575399621431901105389918, −4.34226314442082175722639489901, −3.73584640529047850231900524225, −2.84578598393603317375030722126, −1.57114580344274594324577434867, 1.05837499136676668757913547674, 1.81735691354706522104447335903, 2.68696470654091170819759235469, 3.89711079133647145884411014267, 4.83048845417662975515168496543, 6.30579529087924615571786017595, 6.61174821252443753355718723893, 7.49364865731592233540119768649, 8.281252233865781413946507030074, 8.683493332406301909536050778983

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