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Properties

Label 2-722-19.5-c1-0-10
Degree $2$
Conductor $722$
Sign $0.900 - 0.433i$
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 + (0.222 − 1.26i)3-s + (0.766 − 0.642i)4-s + (2.83 + 2.37i)5-s + (0.222 + 1.26i)6-s + (0.221 + 0.383i)7-s + (−0.500 + 0.866i)8-s + (1.26 + 0.462i)9-s + (−3.47 − 1.26i)10-s + (2.01 − 3.48i)11-s + (−0.642 − 1.11i)12-s + (0.849 + 4.81i)13-s + (−0.338 − 0.284i)14-s + (3.63 − 3.05i)15-s + (0.173 − 0.984i)16-s + (−0.250 + 0.0912i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (0.128 − 0.730i)3-s + (0.383 − 0.321i)4-s + (1.26 + 1.06i)5-s + (0.0910 + 0.516i)6-s + (0.0836 + 0.144i)7-s + (−0.176 + 0.306i)8-s + (0.423 + 0.154i)9-s + (−1.09 − 0.399i)10-s + (0.607 − 1.05i)11-s + (−0.185 − 0.321i)12-s + (0.235 + 1.33i)13-s + (−0.0905 − 0.0760i)14-s + (0.938 − 0.787i)15-s + (0.0434 − 0.246i)16-s + (−0.0608 + 0.0221i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.433i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.900 - 0.433i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ 0.900 - 0.433i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52584 + 0.348364i\)
\(L(\frac12)\) \(\approx\) \(1.52584 + 0.348364i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.222 + 1.26i)T + (-2.81 - 1.02i)T^{2} \)
5 \( 1 + (-2.83 - 2.37i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-0.221 - 0.383i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.01 + 3.48i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.849 - 4.81i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (0.250 - 0.0912i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (7.05 - 5.91i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-0.210 - 0.0764i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (1.73 + 3.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.44T + 37T^{2} \)
41 \( 1 + (1.36 - 7.74i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (4.31 + 3.62i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (2.06 + 0.751i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-7.61 + 6.39i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-3.29 + 1.20i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-3.12 + 2.62i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (0.138 + 0.0505i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-8.78 - 7.37i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-0.246 + 1.40i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-1.77 + 10.0i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-1.64 - 2.84i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.03 + 5.87i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (3.83 - 1.39i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.19318938819263155694505929701, −9.685011290022912594220287249852, −8.781722771462736582745932410174, −7.80797145956426718036927876701, −6.77473790404858904989246587933, −6.42476055880826639896233691002, −5.55247693537391535819196219086, −3.74417649560216236426764204647, −2.25627245910570283834893226794, −1.56766373582321006009361460724, 1.15152072175110170357407724594, 2.29920036740596268846160860525, 3.90994995728677557593108649964, 4.82257591133658236327316353491, 5.81781027981034711502058113700, 6.86776240711759756351829823016, 8.088008853770981333135423975763, 8.902768327816264034837576488773, 9.595166572427502872222416250501, 10.16020726265848958595073676642

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