↑

Properties

Label 2-740-740.287-c1-0-57
Degree $2$
Conductor $740$
Sign $-0.186 + 0.982i$
Analytic cond. $5.90892$
Root an. cond. $2.43082$
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  + (−1.15 − 0.811i)2-s + (0.684 + 1.87i)4-s + (−2.17 + 0.519i)5-s + (0.732 − 2.73i)8-s + (−1.02 + 2.81i)9-s + (2.94 + 1.16i)10-s + (−4.90 − 2.28i)13-s + (−3.06 + 2.57i)16-s + (7.06 − 3.29i)17-s + (3.47 − 2.43i)18-s + (−2.46 − 3.73i)20-s + (4.46 − 2.25i)25-s + (3.83 + 6.63i)26-s + (−4.01 − 6.95i)29-s + (5.63 − 0.493i)32-s + ⋯
L(s)  = 1  + (−0.819 − 0.573i)2-s + (0.342 + 0.939i)4-s + (−0.972 + 0.232i)5-s + (0.258 − 0.965i)8-s + (−0.342 + 0.939i)9-s + (0.930 + 0.367i)10-s + (−1.36 − 0.634i)13-s + (−0.766 + 0.642i)16-s + (1.71 − 0.799i)17-s + (0.819 − 0.573i)18-s + (−0.550 − 0.834i)20-s + (0.892 − 0.451i)25-s + (0.751 + 1.30i)26-s + (−0.745 − 1.29i)29-s + (0.996 − 0.0871i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $-0.186 + 0.982i$
Analytic conductor: \(5.90892\)
Root analytic conductor: \(2.43082\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :1/2),\ -0.186 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.357107 - 0.431209i\)
\(L(\frac12)\) \(\approx\) \(0.357107 - 0.431209i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.15 + 0.811i)T \)
5 \( 1 + (2.17 - 0.519i)T \)
37 \( 1 + (-5.29 + 2.99i)T \)
good3 \( 1 + (1.02 - 2.81i)T^{2} \)
7 \( 1 + (6.89 - 1.21i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.90 + 2.28i)T + (8.35 + 9.95i)T^{2} \)
17 \( 1 + (-7.06 + 3.29i)T + (10.9 - 13.0i)T^{2} \)
19 \( 1 + (17.8 + 6.49i)T^{2} \)
23 \( 1 + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (4.01 + 6.95i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
41 \( 1 + (-11.7 + 4.29i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.26 + 14.5i)T + (-52.1 - 9.20i)T^{2} \)
59 \( 1 + (-10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.81 + 4.97i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (65.9 - 11.6i)T^{2} \)
71 \( 1 + (66.7 + 24.2i)T^{2} \)
73 \( 1 + (7.02 - 7.02i)T - 73iT^{2} \)
79 \( 1 + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (-53.3 + 63.5i)T^{2} \)
89 \( 1 + (-2.01 + 1.68i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (3.84 + 14.3i)T + (-84.0 + 48.5i)T^{2} \)
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

−10.06011120292737490691496746517, −9.521589369707075396459512198916, −8.191140182296077336845379696679, −7.71766872866178397782613112090, −7.21039416244575920491609041994, −5.59382477536803560413911283357, −4.45366329977292066198395414204, −3.23272735198086676590769287701, −2.38455296404750770360171888989, −0.43586906640162340572224103795, 1.16538930715563236393422809886, 2.99763707570021274780286869494, 4.30612666802313543436407907615, 5.41924098150301787805046440913, 6.38718058077528971102940477932, 7.44123510612343512073777996626, 7.85244896606736890829881276354, 8.964724818684473679895287205519, 9.503770865255050967996005272418, 10.44140534360214979312985743932

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