↑

Properties

Label 2-740-740.547-c0-0-0
Degree $2$
Conductor $740$
Sign $0.445 - 0.895i$
Analytic cond. $0.369308$
Root an. cond. $0.607707$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + i·5-s + 0.999i·8-s + (0.866 − 0.5i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.499 + 0.866i)18-s + (0.866 + 0.499i)20-s − 25-s + (−0.366 + 0.366i)29-s + (0.866 + 0.499i)32-s + (−0.866 − 0.499i)34-s − 0.999i·36-s + (0.866 + 0.5i)37-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + i·5-s + 0.999i·8-s + (0.866 − 0.5i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.499 + 0.866i)18-s + (0.866 + 0.499i)20-s − 25-s + (−0.366 + 0.366i)29-s + (0.866 + 0.499i)32-s + (−0.866 − 0.499i)34-s − 0.999i·36-s + (0.866 + 0.5i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $0.445 - 0.895i$
Analytic conductor: \(0.369308\)
Root analytic conductor: \(0.607707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (547, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :0),\ 0.445 - 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6906038458\)
\(L(\frac12)\) \(\approx\) \(0.6906038458\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 - iT \)
37 \( 1 + (-0.866 - 0.5i)T \)
good3 \( 1 + (-0.866 + 0.5i)T^{2} \)
7 \( 1 + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
31 \( 1 - iT^{2} \)
41 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (1.86 - 0.5i)T + (0.866 - 0.5i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 + (-0.866 + 0.5i)T^{2} \)
83 \( 1 + (-0.866 - 0.5i)T^{2} \)
89 \( 1 + (0.5 + 0.133i)T + (0.866 + 0.5i)T^{2} \)
97 \( 1 - 1.73T + T^{2} \)
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

−10.52364472738008781864003336432, −9.865580808505125747071357205473, −9.141986826820556767599086703799, −7.955177587201249068206129938279, −7.36354870971581762264830516389, −6.46479103134850073221741867070, −5.84606518034813986154252965710, −4.35728177321846310550292871526, −3.03133711333930172747107426393, −1.59090044178183410278437293239, 1.15443793449386252972175086115, 2.42304464500207117343682442367, 3.91098326614564103591735517081, 4.81917691633289795937113065316, 6.07679696028216140005878011853, 7.48802752535027932945038149750, 7.76335069770711380815944705751, 8.997046376974766708220561228387, 9.460046933865326854344466571190, 10.31285639807901548577887021552

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