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Properties

Label 2-722-19.4-c1-0-11
Degree $2$
Conductor $722$
Sign $0.290 - 0.956i$
Analytic cond. $5.76519$
Root an. cond. $2.40108$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.520 + 2.95i)3-s + (0.766 + 0.642i)4-s + (1.53 − 1.28i)5-s + (0.520 − 2.95i)6-s + (1.5 − 2.59i)7-s + (−0.500 − 0.866i)8-s + (−5.63 + 2.05i)9-s + (−1.87 + 0.684i)10-s + (1 + 1.73i)11-s + (−1.50 + 2.59i)12-s + (−0.520 + 2.95i)13-s + (−2.29 + 1.92i)14-s + (4.59 + 3.85i)15-s + (0.173 + 0.984i)16-s + (0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (0.300 + 1.70i)3-s + (0.383 + 0.321i)4-s + (0.685 − 0.574i)5-s + (0.212 − 1.20i)6-s + (0.566 − 0.981i)7-s + (−0.176 − 0.306i)8-s + (−1.87 + 0.684i)9-s + (−0.594 + 0.216i)10-s + (0.301 + 0.522i)11-s + (−0.433 + 0.750i)12-s + (−0.144 + 0.819i)13-s + (−0.614 + 0.515i)14-s + (1.18 + 0.995i)15-s + (0.0434 + 0.246i)16-s + (0.227 + 0.0829i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.290 - 0.956i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ 0.290 - 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16361 + 0.862619i\)
\(L(\frac12)\) \(\approx\) \(1.16361 + 0.862619i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 \)
good3 \( 1 + (-0.520 - 2.95i)T + (-2.81 + 1.02i)T^{2} \)
5 \( 1 + (-1.53 + 1.28i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-1.5 + 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.520 - 2.95i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.939 - 0.342i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-3.83 - 3.21i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-2.81 + 1.02i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-3 + 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + (-2.08 - 11.8i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (7.66 - 6.42i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-7.51 + 2.73i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (2.29 + 1.92i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (2.81 + 1.02i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (14.0 - 5.13i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (1.91 + 10.8i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (2.08 + 11.8i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (1 - 1.73i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.04 + 5.90i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (11.2 + 4.10i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.28769857909848278270547159487, −9.633937952928569230507894795790, −9.289694084714025903202998294900, −8.324428260477354545069823851225, −7.37363465543298240596026913292, −6.01773310923540566028370605909, −4.70266448435305778245407445759, −4.33540429127377887640085943468, −3.04448909468035704567638883748, −1.49176080469108518918589333890, 1.02158910664826084112538540231, 2.25723402535977198997959197491, 2.89331966083684186071700091973, 5.35673741728244484946606869934, 6.09431514543459812110587509827, 6.80711034818164631401483282339, 7.63970113353433273782817815966, 8.519780757101468215249149592837, 8.914706649129228347672584686737, 10.23622920548422476260284090209

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