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Properties

Label 2-722-19.18-c2-0-36
Degree $2$
Conductor $722$
Sign $0.506 - 0.862i$
Analytic cond. $19.6730$
Root an. cond. $4.43543$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s + 0.175i·3-s − 2.00·4-s + 7.89·5-s − 0.248·6-s + 7.65·7-s − 2.82i·8-s + 8.96·9-s + 11.1i·10-s − 14.1·11-s − 0.351i·12-s + 6.79i·13-s + 10.8i·14-s + 1.38i·15-s + 4.00·16-s + 17.5·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.0585i·3-s − 0.500·4-s + 1.57·5-s − 0.0414·6-s + 1.09·7-s − 0.353i·8-s + 0.996·9-s + 1.11i·10-s − 1.28·11-s − 0.0292i·12-s + 0.523i·13-s + 0.773i·14-s + 0.0924i·15-s + 0.250·16-s + 1.03·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 - 0.862i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.506 - 0.862i$
Analytic conductor: \(19.6730\)
Root analytic conductor: \(4.43543\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1),\ 0.506 - 0.862i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.758459804\)
\(L(\frac12)\) \(\approx\) \(2.758459804\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
19 \( 1 \)
good3 \( 1 - 0.175iT - 9T^{2} \)
5 \( 1 - 7.89T + 25T^{2} \)
7 \( 1 - 7.65T + 49T^{2} \)
11 \( 1 + 14.1T + 121T^{2} \)
13 \( 1 - 6.79iT - 169T^{2} \)
17 \( 1 - 17.5T + 289T^{2} \)
23 \( 1 - 5.85T + 529T^{2} \)
29 \( 1 + 9.71iT - 841T^{2} \)
31 \( 1 - 12.8iT - 961T^{2} \)
37 \( 1 + 64.6iT - 1.36e3T^{2} \)
41 \( 1 + 61.3iT - 1.68e3T^{2} \)
43 \( 1 + 55.7T + 1.84e3T^{2} \)
47 \( 1 - 48.1T + 2.20e3T^{2} \)
53 \( 1 - 61.4iT - 2.80e3T^{2} \)
59 \( 1 - 70.0iT - 3.48e3T^{2} \)
61 \( 1 - 31.4T + 3.72e3T^{2} \)
67 \( 1 - 87.9iT - 4.48e3T^{2} \)
71 \( 1 - 54.0iT - 5.04e3T^{2} \)
73 \( 1 - 24.8T + 5.32e3T^{2} \)
79 \( 1 - 38.5iT - 6.24e3T^{2} \)
83 \( 1 + 47.5T + 6.88e3T^{2} \)
89 \( 1 - 109. iT - 7.92e3T^{2} \)
97 \( 1 + 42.1iT - 9.40e3T^{2} \)
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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

−10.24455256729249935010944168549, −9.503094995869197103137990973896, −8.607254920020165342601558189902, −7.61333742508512511751384086735, −6.93040685702807787200439009004, −5.59337007415700773986697491907, −5.32579847890579445206108272387, −4.19085617708903731710024045077, −2.40446148020791840544337421068, −1.34092688589175104514509100337, 1.22125275356345180836595106198, 2.03963940466648760707697922608, 3.15999107739884435333644460571, 4.88653170398433135898200367828, 5.19822924564715787073876460991, 6.36992717130114165681599846455, 7.69731059511524779619013896201, 8.343114171765063395478638850978, 9.637053594953587099022538234716, 10.08585905480458787439708734546

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