Properties

Label 2-37-37.36-c5-0-11
Degree $2$
Conductor $37$
Sign $0.937 + 0.347i$
Analytic cond. $5.93420$
Root an. cond. $2.43602$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.04i·2-s + 19.6·3-s + 30.9·4-s − 86.4i·5-s + 20.4i·6-s − 87.1·7-s + 65.5i·8-s + 142.·9-s + 90.1·10-s + 153.·11-s + 606.·12-s + 27.6i·13-s − 90.8i·14-s − 1.69e3i·15-s + 920.·16-s + 926. i·17-s + ⋯
L(s)  = 1  + 0.184i·2-s + 1.25·3-s + 0.966·4-s − 1.54i·5-s + 0.232i·6-s − 0.671·7-s + 0.362i·8-s + 0.585·9-s + 0.285·10-s + 0.382·11-s + 1.21·12-s + 0.0453i·13-s − 0.123i·14-s − 1.94i·15-s + 0.899·16-s + 0.777i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.937 + 0.347i$
Analytic conductor: \(5.93420\)
Root analytic conductor: \(2.43602\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :5/2),\ 0.937 + 0.347i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.52620 - 0.452816i\)
\(L(\frac12)\) \(\approx\) \(2.52620 - 0.452816i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (7.80e3 + 2.89e3i)T \)
good2 \( 1 - 1.04iT - 32T^{2} \)
3 \( 1 - 19.6T + 243T^{2} \)
5 \( 1 + 86.4iT - 3.12e3T^{2} \)
7 \( 1 + 87.1T + 1.68e4T^{2} \)
11 \( 1 - 153.T + 1.61e5T^{2} \)
13 \( 1 - 27.6iT - 3.71e5T^{2} \)
17 \( 1 - 926. iT - 1.41e6T^{2} \)
19 \( 1 - 2.74e3iT - 2.47e6T^{2} \)
23 \( 1 - 1.33e3iT - 6.43e6T^{2} \)
29 \( 1 + 7.14e3iT - 2.05e7T^{2} \)
31 \( 1 - 4.02e3iT - 2.86e7T^{2} \)
41 \( 1 - 1.13e4T + 1.15e8T^{2} \)
43 \( 1 + 8.18e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.33e4T + 2.29e8T^{2} \)
53 \( 1 - 2.76e4T + 4.18e8T^{2} \)
59 \( 1 - 1.60e4iT - 7.14e8T^{2} \)
61 \( 1 + 4.49e4iT - 8.44e8T^{2} \)
67 \( 1 + 4.58e4T + 1.35e9T^{2} \)
71 \( 1 + 1.87e4T + 1.80e9T^{2} \)
73 \( 1 + 6.61e3T + 2.07e9T^{2} \)
79 \( 1 - 6.99e4iT - 3.07e9T^{2} \)
83 \( 1 - 8.13e4T + 3.93e9T^{2} \)
89 \( 1 + 1.20e5iT - 5.58e9T^{2} \)
97 \( 1 - 5.97e3iT - 8.58e9T^{2} \)
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

−15.38175887470251880237560815510, −14.18803910688171256264157522821, −12.91535718371616109329369336919, −11.97516213453424221769628124994, −9.962220135440288680876350344480, −8.732579246246036235776633847713, −7.79733851870874441915187857129, −5.91510436496787463521400357817, −3.68578957799826586652423846780, −1.75204905267410068293627041215, 2.55187196992590657976890863887, 3.24979136544796284429555512764, 6.58299571213017934262285619074, 7.37898920660802353206823544912, 9.197919278997528170366076317334, 10.49799872862188980130037156322, 11.56722442870132040111722750430, 13.27952778199272704376451983145, 14.48677787822871186220517566752, 15.13400128519610299455431155701

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