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Properties

Label 2-722-19.5-c1-0-1
Degree $2$
Conductor $722$
Sign $-0.527 - 0.849i$
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 + (−0.173 + 0.984i)3-s + (0.766 − 0.642i)4-s + (−3.06 − 2.57i)5-s + (−0.173 − 0.984i)6-s + (−1.5 − 2.59i)7-s + (−0.500 + 0.866i)8-s + (1.87 + 0.684i)9-s + (3.75 + 1.36i)10-s + (−1 + 1.73i)11-s + (0.5 + 0.866i)12-s + (−0.173 − 0.984i)13-s + (2.29 + 1.92i)14-s + (3.06 − 2.57i)15-s + (0.173 − 0.984i)16-s + (−2.81 + 1.02i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (−0.100 + 0.568i)3-s + (0.383 − 0.321i)4-s + (−1.37 − 1.14i)5-s + (−0.0708 − 0.402i)6-s + (−0.566 − 0.981i)7-s + (−0.176 + 0.306i)8-s + (0.626 + 0.228i)9-s + (1.18 + 0.432i)10-s + (−0.301 + 0.522i)11-s + (0.144 + 0.249i)12-s + (−0.0481 − 0.273i)13-s + (0.614 + 0.515i)14-s + (0.791 − 0.663i)15-s + (0.0434 − 0.246i)16-s + (−0.683 + 0.248i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.527 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $-0.527 - 0.849i$
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: \((2,\ 722,\ (\ :1/2),\ -0.527 - 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.158696 + 0.285186i\)
\(L(\frac12)\) \(\approx\) \(0.158696 + 0.285186i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.173 - 0.984i)T + (-2.81 - 1.02i)T^{2} \)
5 \( 1 + (3.06 + 2.57i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.173 + 0.984i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (2.81 - 1.02i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (0.766 - 0.642i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-4.69 - 1.71i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (1.38 - 7.87i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-3.06 - 2.57i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (7.51 + 2.73i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (0.766 - 0.642i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (14.0 - 5.13i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-1.53 + 1.28i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (2.81 + 1.02i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-1.53 - 1.28i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-1.56 + 8.86i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (1.73 - 9.84i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-3 - 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-1.87 + 0.684i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.49183855932811632542537078354, −9.884032557297493705445812947548, −8.940789425554322974310389130719, −8.102101725376873178279771980129, −7.43102199971826631394438286744, −6.60204532584448145305420509757, −4.91724642523211689240275911233, −4.46674233038011955136639474347, −3.40347053322413799094085903922, −1.21633162996391308057239264654, 0.23923187676243014600849423162, 2.32572427428437487014186549568, 3.21197856846635753144969647863, 4.31393606191331097456211681659, 6.15080260612543762168183615104, 6.73428907292662787070332254243, 7.56669585204420472059498762514, 8.260758282525583124551017300562, 9.227247316503088433841470973553, 10.21133709863887452497391280312

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