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Properties

Label 2-722-19.6-c1-0-23
Degree $2$
Conductor $722$
Sign $-0.0225 + 0.999i$
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  + (0.766 − 0.642i)2-s + (0.939 + 0.342i)3-s + (0.173 − 0.984i)4-s + (0.939 − 0.342i)6-s + (2 − 3.46i)7-s + (−0.500 − 0.866i)8-s + (−1.53 − 1.28i)9-s + (−1.5 − 2.59i)11-s + (0.499 − 0.866i)12-s + (1.87 − 0.684i)13-s + (−0.694 − 3.93i)14-s + (−0.939 − 0.342i)16-s + (−4.59 + 3.85i)17-s − 2·18-s + (3.06 − 2.57i)21-s + (−2.81 − 1.02i)22-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (0.542 + 0.197i)3-s + (0.0868 − 0.492i)4-s + (0.383 − 0.139i)6-s + (0.755 − 1.30i)7-s + (−0.176 − 0.306i)8-s + (−0.510 − 0.428i)9-s + (−0.452 − 0.783i)11-s + (0.144 − 0.250i)12-s + (0.521 − 0.189i)13-s + (−0.185 − 1.05i)14-s + (−0.234 − 0.0855i)16-s + (−1.11 + 0.935i)17-s − 0.471·18-s + (0.668 − 0.561i)21-s + (−0.601 − 0.218i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0225 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0225 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $-0.0225 + 0.999i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (595, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ -0.0225 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64714 - 1.68464i\)
\(L(\frac12)\) \(\approx\) \(1.64714 - 1.68464i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 \)
good3 \( 1 + (-0.939 - 0.342i)T + (2.29 + 1.92i)T^{2} \)
5 \( 1 + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-2 + 3.46i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.87 + 0.684i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (4.59 - 3.85i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (1.04 - 5.90i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + (-8.45 - 3.07i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.694 + 3.93i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (1.04 - 5.90i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-6.89 + 5.78i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.694 - 3.93i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-5.36 - 4.49i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-1.04 - 5.90i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-0.939 - 0.342i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (3.75 + 1.36i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (1.5 - 2.59i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.63 + 2.05i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (13.0 - 10.9i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.45695351776810060805113361596, −9.390454993807584052143060143976, −8.437679171099517064723401687136, −7.76791960431311512268667599258, −6.51703171935528379671907962451, −5.61510673657310915259443832053, −4.32562427520217002642406007901, −3.73794623536744336668601287618, −2.58811909730549232157672224177, −1.01083604005651643567577493704, 2.21075371326934445856806379838, 2.76386385495326814592204699271, 4.48357259861254960198815730149, 5.13496683457295790015460276343, 6.12874376063575851997725525380, 7.15901598946654611560316252680, 8.129224899610128774825051671988, 8.673781431302691713305786089665, 9.420697362061977946514879846952, 10.93431114496307178752569832214

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