↑

Properties

Label 2-37-37.36-c3-0-0
Degree $2$
Conductor $37$
Sign $-0.891 + 0.452i$
Analytic cond. $2.18307$
Root an. cond. $1.47752$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.73i·2-s − 7.61·3-s − 5.96·4-s − 2.08i·5-s − 28.4i·6-s − 23.3·7-s + 7.60i·8-s + 31.0·9-s + 7.80·10-s − 2.84·11-s + 45.4·12-s + 70.5i·13-s − 87.3i·14-s + 15.9i·15-s − 76.1·16-s + 76.2i·17-s + ⋯
L(s)  = 1  + 1.32i·2-s − 1.46·3-s − 0.745·4-s − 0.186i·5-s − 1.93i·6-s − 1.26·7-s + 0.336i·8-s + 1.14·9-s + 0.246·10-s − 0.0778·11-s + 1.09·12-s + 1.50i·13-s − 1.66i·14-s + 0.273i·15-s − 1.18·16-s + 1.08i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.891 + 0.452i$
Analytic conductor: \(2.18307\)
Root analytic conductor: \(1.47752\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :3/2),\ -0.891 + 0.452i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0999497 - 0.418181i\)
\(L(\frac12)\) \(\approx\) \(0.0999497 - 0.418181i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (200. - 101. i)T \)
good2 \( 1 - 3.73iT - 8T^{2} \)
3 \( 1 + 7.61T + 27T^{2} \)
5 \( 1 + 2.08iT - 125T^{2} \)
7 \( 1 + 23.3T + 343T^{2} \)
11 \( 1 + 2.84T + 1.33e3T^{2} \)
13 \( 1 - 70.5iT - 2.19e3T^{2} \)
17 \( 1 - 76.2iT - 4.91e3T^{2} \)
19 \( 1 + 22.4iT - 6.85e3T^{2} \)
23 \( 1 + 159. iT - 1.21e4T^{2} \)
29 \( 1 + 126. iT - 2.43e4T^{2} \)
31 \( 1 - 272. iT - 2.97e4T^{2} \)
41 \( 1 - 9.53T + 6.89e4T^{2} \)
43 \( 1 - 286. iT - 7.95e4T^{2} \)
47 \( 1 + 580.T + 1.03e5T^{2} \)
53 \( 1 - 493.T + 1.48e5T^{2} \)
59 \( 1 - 456. iT - 2.05e5T^{2} \)
61 \( 1 + 156. iT - 2.26e5T^{2} \)
67 \( 1 - 264.T + 3.00e5T^{2} \)
71 \( 1 - 99.7T + 3.57e5T^{2} \)
73 \( 1 + 471.T + 3.89e5T^{2} \)
79 \( 1 + 559. iT - 4.93e5T^{2} \)
83 \( 1 - 694.T + 5.71e5T^{2} \)
89 \( 1 + 351. iT - 7.04e5T^{2} \)
97 \( 1 + 543. iT - 9.12e5T^{2} \)
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

−16.48677238184024067244949033195, −15.97678291898002843431146163112, −14.52559208459044178397234771784, −12.98666761855898563702925170318, −11.82983304064805195357750837408, −10.44601408230244806836725440199, −8.783728897312420115410638211187, −6.71529562665765827338132586161, −6.33368593320922421835413289436, −4.77915149559462255445209375472, 0.42688964957054755265736946320, 3.25631944161266833888874095101, 5.46073927816365930619851125771, 6.93528086878899315201948435406, 9.648679769383215046276848117089, 10.51388637241653996802149760934, 11.49346435179736605513339724081, 12.49994084332655004637083988634, 13.26335209717160586818808556506, 15.57584952813157126295237790407

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