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Properties

Label 2-723-723.236-c0-0-0
Degree $2$
Conductor $723$
Sign $-0.991 - 0.128i$
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  + (−0.453 + 0.891i)3-s + i·4-s + (−1.93 − 0.465i)7-s + (−0.587 − 0.809i)9-s + (−0.891 − 0.453i)12-s + (−0.119 + 1.51i)13-s − 16-s + (0.0600 + 0.144i)19-s + (1.29 − 1.51i)21-s + (−0.587 + 0.809i)25-s + (0.987 − 0.156i)27-s + (0.465 − 1.93i)28-s + (−0.965 − 1.57i)31-s + (0.809 − 0.587i)36-s + (0.0366 + 0.465i)37-s + ⋯
L(s)  = 1  + (−0.453 + 0.891i)3-s + i·4-s + (−1.93 − 0.465i)7-s + (−0.587 − 0.809i)9-s + (−0.891 − 0.453i)12-s + (−0.119 + 1.51i)13-s − 16-s + (0.0600 + 0.144i)19-s + (1.29 − 1.51i)21-s + (−0.587 + 0.809i)25-s + (0.987 − 0.156i)27-s + (0.465 − 1.93i)28-s + (−0.965 − 1.57i)31-s + (0.809 − 0.587i)36-s + (0.0366 + 0.465i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(723\)    =    \(3 \cdot 241\)
Sign: $-0.991 - 0.128i$
Analytic conductor: \(0.360824\)
Root analytic conductor: \(0.600686\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{723} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 723,\ (\ :0),\ -0.991 - 0.128i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4205431287\)
\(L(\frac12)\) \(\approx\) \(0.4205431287\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.453 - 0.891i)T \)
241 \( 1 + (0.453 - 0.891i)T \)
good2 \( 1 - iT^{2} \)
5 \( 1 + (0.587 - 0.809i)T^{2} \)
7 \( 1 + (1.93 + 0.465i)T + (0.891 + 0.453i)T^{2} \)
11 \( 1 + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + (0.119 - 1.51i)T + (-0.987 - 0.156i)T^{2} \)
17 \( 1 + (0.453 - 0.891i)T^{2} \)
19 \( 1 + (-0.0600 - 0.144i)T + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + (0.156 - 0.987i)T^{2} \)
29 \( 1 + (0.951 + 0.309i)T^{2} \)
31 \( 1 + (0.965 + 1.57i)T + (-0.453 + 0.891i)T^{2} \)
37 \( 1 + (-0.0366 - 0.465i)T + (-0.987 + 0.156i)T^{2} \)
41 \( 1 + (0.951 - 0.309i)T^{2} \)
43 \( 1 + (-0.398 - 1.65i)T + (-0.891 + 0.453i)T^{2} \)
47 \( 1 + (0.951 + 0.309i)T^{2} \)
53 \( 1 + (-0.951 - 0.309i)T^{2} \)
59 \( 1 + (-0.587 + 0.809i)T^{2} \)
61 \( 1 + (-1.04 - 0.533i)T + (0.587 + 0.809i)T^{2} \)
67 \( 1 + (-0.297 - 1.87i)T + (-0.951 + 0.309i)T^{2} \)
71 \( 1 + (-0.891 + 0.453i)T^{2} \)
73 \( 1 + (-0.0819 - 1.04i)T + (-0.987 + 0.156i)T^{2} \)
79 \( 1 + (0.550 - 0.280i)T + (0.587 - 0.809i)T^{2} \)
83 \( 1 + (-0.809 - 0.587i)T^{2} \)
89 \( 1 + (-0.707 - 0.707i)T^{2} \)
97 \( 1 + (0.533 + 0.734i)T + (-0.309 + 0.951i)T^{2} \)
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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

−11.17326745169593905818354951867, −9.811715058547117723321851304841, −9.574884225182431279688755663894, −8.734865877196723599339775893265, −7.34658058704516249353294048967, −6.65911143372720723350196383126, −5.80482490461680765053179889217, −4.23117183244589962537653054605, −3.80449979164372150226746568759, −2.76734462571796161873970638185, 0.44872896704561440138061203752, 2.26700261175387134565059522331, 3.36607628192127862562593292370, 5.23599477423343332307347952069, 5.82861343527542437067375403375, 6.53811899317011842916178293727, 7.29271619954825355590801948819, 8.562890955844853456320439362604, 9.468593708422713064440964077408, 10.31668191551556392593873615014

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