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Properties

Label 2-722-19.4-c1-0-10
Degree $2$
Conductor $722$
Sign $0.920 - 0.391i$
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 + (0.0768 + 0.435i)3-s + (0.766 + 0.642i)4-s + (−0.682 + 0.573i)5-s + (0.0768 − 0.435i)6-s + (−1.26 + 2.18i)7-s + (−0.500 − 0.866i)8-s + (2.63 − 0.959i)9-s + (0.837 − 0.304i)10-s + (−0.975 − 1.69i)11-s + (−0.221 + 0.383i)12-s + (1.12 − 6.36i)13-s + (1.93 − 1.61i)14-s + (−0.302 − 0.253i)15-s + (0.173 + 0.984i)16-s + (3.21 + 1.17i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (0.0443 + 0.251i)3-s + (0.383 + 0.321i)4-s + (−0.305 + 0.256i)5-s + (0.0313 − 0.177i)6-s + (−0.476 + 0.824i)7-s + (−0.176 − 0.306i)8-s + (0.878 − 0.319i)9-s + (0.264 − 0.0964i)10-s + (−0.294 − 0.509i)11-s + (−0.0638 + 0.110i)12-s + (0.311 − 1.76i)13-s + (0.515 − 0.432i)14-s + (−0.0780 − 0.0654i)15-s + (0.0434 + 0.246i)16-s + (0.779 + 0.283i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.920 - 0.391i$
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: \((2,\ 722,\ (\ :1/2),\ 0.920 - 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06737 + 0.217741i\)
\(L(\frac12)\) \(\approx\) \(1.06737 + 0.217741i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0768 - 0.435i)T + (-2.81 + 1.02i)T^{2} \)
5 \( 1 + (0.682 - 0.573i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (1.26 - 2.18i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.975 + 1.69i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.12 + 6.36i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-3.21 - 1.17i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-6.27 - 5.26i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-4.30 + 1.56i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (4.39 - 7.61i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.97T + 37T^{2} \)
41 \( 1 + (-0.605 - 3.43i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (4.78 - 4.01i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-9.91 + 3.60i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (2.88 + 2.41i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-2.67 - 0.971i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (1.88 + 1.57i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (2.51 - 0.914i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (0.0432 - 0.0362i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (1.21 + 6.86i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-1.58 - 8.99i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-7.06 + 12.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.324 - 1.83i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-7.35 - 2.67i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.41945310063074979366192915990, −9.633361383995167122745113937331, −8.864342614052733844399896791229, −7.938755845680370954927398502049, −7.21237269793777915641765040106, −6.03058426312769993914025448470, −5.17851156482120355783209583326, −3.45319672256222038767865358719, −3.01344543642675877747669186679, −1.12241875043723337099430558629, 0.916299763588230930057774121623, 2.29362644664475799152998138982, 4.00534337117960423842017935249, 4.75067274314951499612831312926, 6.30396351093338856336322741555, 7.08225513383396972958921144478, 7.55800134252981304633995132178, 8.662792708655049092717179600769, 9.502374030480339656723236428335, 10.22334809097957462208416651929

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