Properties

Label 2-722-19.11-c1-0-24
Degree $2$
Conductor $722$
Sign $-0.919 + 0.392i$
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  + (0.5 − 0.866i)2-s + (1.39 − 2.41i)3-s + (−0.499 − 0.866i)4-s + (1.17 − 2.03i)5-s + (−1.39 − 2.41i)6-s − 1.28·7-s − 0.999·8-s + (−2.40 − 4.16i)9-s + (−1.17 − 2.03i)10-s + 5.75·11-s − 2.79·12-s + (0.152 + 0.263i)13-s + (−0.642 + 1.11i)14-s + (−3.27 − 5.67i)15-s + (−0.5 + 0.866i)16-s + (−2.09 + 3.62i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.806 − 1.39i)3-s + (−0.249 − 0.433i)4-s + (0.524 − 0.908i)5-s + (−0.570 − 0.987i)6-s − 0.485·7-s − 0.353·8-s + (−0.800 − 1.38i)9-s + (−0.370 − 0.642i)10-s + 1.73·11-s − 0.806·12-s + (0.0421 + 0.0730i)13-s + (−0.171 + 0.297i)14-s + (−0.845 − 1.46i)15-s + (−0.125 + 0.216i)16-s + (−0.507 + 0.879i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 + 0.392i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.919 + 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $-0.919 + 0.392i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (429, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ -0.919 + 0.392i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.499895 - 2.44632i\)
\(L(\frac12)\) \(\approx\) \(0.499895 - 2.44632i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 \)
good3 \( 1 + (-1.39 + 2.41i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.17 + 2.03i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 1.28T + 7T^{2} \)
11 \( 1 - 5.75T + 11T^{2} \)
13 \( 1 + (-0.152 - 0.263i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.09 - 3.62i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.23 - 5.60i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.56 + 2.70i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.44T + 31T^{2} \)
37 \( 1 - 3.97T + 37T^{2} \)
41 \( 1 + (2.50 - 4.34i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.494 + 0.856i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.19 - 3.80i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.64 - 2.84i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.65 + 2.86i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.48 - 9.49i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.19 + 3.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.20 - 3.82i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.13 + 1.96i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.28 + 7.42i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.76T + 83T^{2} \)
89 \( 1 + (7.53 + 13.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.67 + 15.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.703523750553093649846893862778, −9.115452743674684490784678919985, −8.580811569804229600764950027010, −7.35356857289574644908109634907, −6.47874781084455953557646522810, −5.71422613927125185130724752737, −4.22797302718528874745357430341, −3.22486513871458333603141495639, −1.82814389955525570777619100991, −1.22915890725299477097686468502, 2.54666346005281100741003237578, 3.48476402849334215660787269892, 4.21424318359123275379265760124, 5.28535036392834032139692578931, 6.53227337898396673969526046151, 6.98422622664233916650327222050, 8.468870670439075953545954059456, 9.214963721153592668006623668263, 9.632364418418509836859569158191, 10.61808582668024972637802474476

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