Properties

Label 2-737-737.516-c0-0-0
Degree $2$
Conductor $737$
Sign $-0.0864 + 0.996i$
Analytic cond. $0.367810$
Root an. cond. $0.606474$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.273 − 1.89i)3-s + (0.0475 − 0.998i)4-s + (0.815 + 1.78i)5-s + (−2.57 − 0.755i)9-s + (0.580 − 0.814i)11-s + (−1.88 − 0.363i)12-s + (3.61 − 1.06i)15-s + (−0.995 − 0.0950i)16-s + (1.82 − 0.729i)20-s + (0.223 + 0.175i)23-s + (−1.87 + 2.15i)25-s + (−1.34 + 2.93i)27-s + (−0.154 + 0.445i)31-s + (−1.38 − 1.32i)33-s + (−0.877 + 2.53i)36-s + (−0.235 + 0.408i)37-s + ⋯
L(s)  = 1  + (0.273 − 1.89i)3-s + (0.0475 − 0.998i)4-s + (0.815 + 1.78i)5-s + (−2.57 − 0.755i)9-s + (0.580 − 0.814i)11-s + (−1.88 − 0.363i)12-s + (3.61 − 1.06i)15-s + (−0.995 − 0.0950i)16-s + (1.82 − 0.729i)20-s + (0.223 + 0.175i)23-s + (−1.87 + 2.15i)25-s + (−1.34 + 2.93i)27-s + (−0.154 + 0.445i)31-s + (−1.38 − 1.32i)33-s + (−0.877 + 2.53i)36-s + (−0.235 + 0.408i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0864 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0864 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(737\)    =    \(11 \cdot 67\)
Sign: $-0.0864 + 0.996i$
Analytic conductor: \(0.367810\)
Root analytic conductor: \(0.606474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{737} (516, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 737,\ (\ :0),\ -0.0864 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.136872882\)
\(L(\frac12)\) \(\approx\) \(1.136872882\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.580 + 0.814i)T \)
67 \( 1 + (0.786 - 0.618i)T \)
good2 \( 1 + (-0.0475 + 0.998i)T^{2} \)
3 \( 1 + (-0.273 + 1.89i)T + (-0.959 - 0.281i)T^{2} \)
5 \( 1 + (-0.815 - 1.78i)T + (-0.654 + 0.755i)T^{2} \)
7 \( 1 + (0.888 - 0.458i)T^{2} \)
13 \( 1 + (0.786 - 0.618i)T^{2} \)
17 \( 1 + (0.995 - 0.0950i)T^{2} \)
19 \( 1 + (0.888 + 0.458i)T^{2} \)
23 \( 1 + (-0.223 - 0.175i)T + (0.235 + 0.971i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.154 - 0.445i)T + (-0.786 - 0.618i)T^{2} \)
37 \( 1 + (0.235 - 0.408i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.580 - 0.814i)T^{2} \)
43 \( 1 + (-0.415 + 0.909i)T^{2} \)
47 \( 1 + (-1.34 + 0.537i)T + (0.723 - 0.690i)T^{2} \)
53 \( 1 + (-0.975 - 0.627i)T + (0.415 + 0.909i)T^{2} \)
59 \( 1 + (0.759 + 0.876i)T + (-0.142 + 0.989i)T^{2} \)
61 \( 1 + (0.327 - 0.945i)T^{2} \)
71 \( 1 + (0.0623 - 1.30i)T + (-0.995 - 0.0950i)T^{2} \)
73 \( 1 + (0.327 - 0.945i)T^{2} \)
79 \( 1 + (-0.928 - 0.371i)T^{2} \)
83 \( 1 + (-0.981 - 0.189i)T^{2} \)
89 \( 1 + (-0.0930 - 0.647i)T + (-0.959 + 0.281i)T^{2} \)
97 \( 1 + (-0.888 + 1.53i)T + (-0.5 - 0.866i)T^{2} \)
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

−10.55020750085432576805836242215, −9.496769288541104578884292693578, −8.588553609663285133042470705634, −7.31567692341142390348814880396, −6.85435880689376011476242196160, −6.10713538012702436465293891994, −5.68221530856320317736527319727, −3.25249219204623532498490726619, −2.38212142008290081637571970405, −1.41648548041846735607893900870, 2.26102553418900920314007930962, 3.70137043696885235813030736481, 4.47667822317165963391544052713, 4.99404312003872503299989877462, 6.04859060975490957397515336242, 7.75502975142203287993084973474, 8.735349867238403449679347160751, 9.082508517966461716786051196484, 9.680593269140965115636350259948, 10.54889187935788592697953801071

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