Properties

Label 2-722-19.5-c1-0-0
Degree $2$
Conductor $722$
Sign $-0.281 - 0.959i$
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 + (0.485 − 2.75i)3-s + (0.766 − 0.642i)4-s + (−1.79 − 1.50i)5-s + (0.485 + 2.75i)6-s + (0.642 + 1.11i)7-s + (−0.500 + 0.866i)8-s + (−4.51 − 1.64i)9-s + (2.20 + 0.801i)10-s + (−2.87 + 4.98i)11-s + (−1.39 − 2.41i)12-s + (0.0528 + 0.299i)13-s + (−0.983 − 0.825i)14-s + (−5.01 + 4.21i)15-s + (0.173 − 0.984i)16-s + (−3.93 + 1.43i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (0.280 − 1.58i)3-s + (0.383 − 0.321i)4-s + (−0.803 − 0.673i)5-s + (0.198 + 1.12i)6-s + (0.242 + 0.420i)7-s + (−0.176 + 0.306i)8-s + (−1.50 − 0.547i)9-s + (0.696 + 0.253i)10-s + (−0.867 + 1.50i)11-s + (−0.403 − 0.698i)12-s + (0.0146 + 0.0831i)13-s + (−0.262 − 0.220i)14-s + (−1.29 + 1.08i)15-s + (0.0434 − 0.246i)16-s + (−0.954 + 0.347i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $-0.281 - 0.959i$
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: \((2,\ 722,\ (\ :1/2),\ -0.281 - 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0356241 + 0.0475842i\)
\(L(\frac12)\) \(\approx\) \(0.0356241 + 0.0475842i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.485 + 2.75i)T + (-2.81 - 1.02i)T^{2} \)
5 \( 1 + (1.79 + 1.50i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-0.642 - 1.11i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.87 - 4.98i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.0528 - 0.299i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (3.93 - 1.43i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (4.96 - 4.16i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-2.93 - 1.06i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (3.22 + 5.57i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.97T + 37T^{2} \)
41 \( 1 + (0.871 - 4.94i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (0.757 + 0.635i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-4.12 - 1.50i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-2.52 + 2.11i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (3.11 - 1.13i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (8.39 - 7.04i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-4.11 - 1.49i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (3.38 + 2.83i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (0.393 - 2.23i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-1.48 + 8.44i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-4.88 - 8.45i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.61 + 14.8i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (16.3 - 5.93i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.66016368960707772577809127211, −9.533366707969648531177267764212, −8.588450415075703775732541807072, −7.961780941937626088078750182219, −7.44994833172846980233769873856, −6.63685580232745195144864647947, −5.51733784212653874112178944836, −4.31371537067001861890836803651, −2.43517192385054151910659008974, −1.63880664823946982169124090879, 0.03518258495033353923458803141, 2.71306603554090351487367363830, 3.50615254352846147202254980017, 4.33135785945573145672596398487, 5.50078365762029883313384732712, 6.80672409886032364096099762473, 7.946001853763672742888052742165, 8.533795679248643692830884897865, 9.314415518808728531562983716005, 10.54832128200615291331725708627

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