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Properties

Label 2-722-19.9-c1-0-10
Degree $2$
Conductor $722$
Sign $0.296 - 0.955i$
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  + (0.173 + 0.984i)2-s + (0.983 − 0.825i)3-s + (−0.939 + 0.342i)4-s + (−3.47 − 1.26i)5-s + (0.983 + 0.825i)6-s + (0.221 − 0.383i)7-s + (−0.5 − 0.866i)8-s + (−0.234 + 1.33i)9-s + (0.641 − 3.63i)10-s + (2.01 + 3.48i)11-s + (−0.642 + 1.11i)12-s + (3.74 + 3.14i)13-s + (0.415 + 0.151i)14-s + (−4.45 + 1.62i)15-s + (0.766 − 0.642i)16-s + (0.0463 + 0.262i)17-s + ⋯
L(s)  = 1  + (0.122 + 0.696i)2-s + (0.567 − 0.476i)3-s + (−0.469 + 0.171i)4-s + (−1.55 − 0.565i)5-s + (0.401 + 0.336i)6-s + (0.0836 − 0.144i)7-s + (−0.176 − 0.306i)8-s + (−0.0782 + 0.443i)9-s + (0.202 − 1.15i)10-s + (0.607 + 1.05i)11-s + (−0.185 + 0.321i)12-s + (1.03 + 0.872i)13-s + (0.111 + 0.0404i)14-s + (−1.15 + 0.419i)15-s + (0.191 − 0.160i)16-s + (0.0112 + 0.0637i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.296 - 0.955i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ 0.296 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14029 + 0.840247i\)
\(L(\frac12)\) \(\approx\) \(1.14029 + 0.840247i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 \)
good3 \( 1 + (-0.983 + 0.825i)T + (0.520 - 2.95i)T^{2} \)
5 \( 1 + (3.47 + 1.26i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-0.221 + 0.383i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.01 - 3.48i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.74 - 3.14i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.0463 - 0.262i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-8.65 + 3.14i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.0388 - 0.220i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (1.73 - 3.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.44T + 37T^{2} \)
41 \( 1 + (6.02 - 5.05i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-5.29 - 1.92i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.381 + 2.16i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (9.34 - 3.40i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (0.609 + 3.45i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (3.83 - 1.39i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.0256 + 0.145i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (10.7 + 3.92i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-1.08 + 0.914i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-7.81 + 6.55i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-1.64 + 2.84i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.56 + 3.83i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-0.709 - 4.02i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.74431376240575812281210555250, −9.135131133376895829952180252649, −8.752869332453460004308817732845, −7.85632012478255865150821271192, −7.27052134231216334578658797135, −6.51585963609099087202538486013, −4.86005138865167089498872989748, −4.33806686603672842515163373213, −3.23563430699650815713235180306, −1.37462336301577599028008828585, 0.78566396627995879363134002800, 3.08843082738602545776778225967, 3.42775984082986075024790837140, 4.20953777667054336420671906851, 5.61172061000398819630385479833, 6.79340081044646493244699926366, 7.931118617318502376443008263545, 8.654012088418148725944905650841, 9.235751389495894138883363748049, 10.50734448355061122619900769332

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