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Properties

Label 2-723-723.509-c0-0-0
Degree $2$
Conductor $723$
Sign $-0.0234 - 0.999i$
Analytic cond. $0.360824$
Root an. cond. $0.600686$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.987 − 0.156i)3-s + i·4-s + (0.303 + 0.355i)7-s + (0.951 + 0.309i)9-s + (0.156 − 0.987i)12-s + (−1.65 + 0.398i)13-s − 16-s + (0.744 + 1.79i)19-s + (−0.243 − 0.398i)21-s + (0.951 − 0.309i)25-s + (−0.891 − 0.453i)27-s + (−0.355 + 0.303i)28-s + (−0.144 + 1.84i)31-s + (−0.309 + 0.951i)36-s + (−1.47 − 0.355i)37-s + ⋯
L(s)  = 1  + (−0.987 − 0.156i)3-s + i·4-s + (0.303 + 0.355i)7-s + (0.951 + 0.309i)9-s + (0.156 − 0.987i)12-s + (−1.65 + 0.398i)13-s − 16-s + (0.744 + 1.79i)19-s + (−0.243 − 0.398i)21-s + (0.951 − 0.309i)25-s + (−0.891 − 0.453i)27-s + (−0.355 + 0.303i)28-s + (−0.144 + 1.84i)31-s + (−0.309 + 0.951i)36-s + (−1.47 − 0.355i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0234 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0234 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(723\)    =    \(3 \cdot 241\)
Sign: $-0.0234 - 0.999i$
Analytic conductor: \(0.360824\)
Root analytic conductor: \(0.600686\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{723} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 723,\ (\ :0),\ -0.0234 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6301650336\)
\(L(\frac12)\) \(\approx\) \(0.6301650336\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.987 + 0.156i)T \)
241 \( 1 + (0.987 + 0.156i)T \)
good2 \( 1 - iT^{2} \)
5 \( 1 + (-0.951 + 0.309i)T^{2} \)
7 \( 1 + (-0.303 - 0.355i)T + (-0.156 + 0.987i)T^{2} \)
11 \( 1 + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + (1.65 - 0.398i)T + (0.891 - 0.453i)T^{2} \)
17 \( 1 + (0.987 + 0.156i)T^{2} \)
19 \( 1 + (-0.744 - 1.79i)T + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + (0.453 + 0.891i)T^{2} \)
29 \( 1 + (0.587 - 0.809i)T^{2} \)
31 \( 1 + (0.144 - 1.84i)T + (-0.987 - 0.156i)T^{2} \)
37 \( 1 + (1.47 + 0.355i)T + (0.891 + 0.453i)T^{2} \)
41 \( 1 + (0.587 + 0.809i)T^{2} \)
43 \( 1 + (-1.51 - 1.29i)T + (0.156 + 0.987i)T^{2} \)
47 \( 1 + (0.587 - 0.809i)T^{2} \)
53 \( 1 + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (0.951 - 0.309i)T^{2} \)
61 \( 1 + (-0.297 + 1.87i)T + (-0.951 - 0.309i)T^{2} \)
67 \( 1 + (-0.533 + 1.04i)T + (-0.587 - 0.809i)T^{2} \)
71 \( 1 + (0.156 + 0.987i)T^{2} \)
73 \( 1 + (-0.152 - 0.0366i)T + (0.891 + 0.453i)T^{2} \)
79 \( 1 + (0.253 + 1.59i)T + (-0.951 + 0.309i)T^{2} \)
83 \( 1 + (0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.707 - 0.707i)T^{2} \)
97 \( 1 + (-1.87 - 0.610i)T + (0.809 + 0.587i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.92431285383144784373513226354, −10.08388114150285946738223519172, −9.141237580273906941690038543741, −8.034671361599291048836044320479, −7.33648262356390816309465327607, −6.57182251716174475703552990467, −5.30748527604065035567967834627, −4.64494126895872151057203693844, −3.36905028940241606625221798933, −1.93994628129376184274284164000, 0.78153479464684803768118330561, 2.46536572948297363872433937051, 4.35089709019206006531495640942, 5.08761399257753168753367762041, 5.69286999902279252867280911198, 7.00585923398534502619990105843, 7.31475408027262089119783231932, 9.023142479802125125196277958734, 9.744490860163509576863816790714, 10.44470635895495390404694722860

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