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Properties

Label 2-722-19.9-c1-0-20
Degree $2$
Conductor $722$
Sign $0.910 + 0.413i$
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 + (−0.766 + 0.642i)3-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)6-s + (2 − 3.46i)7-s + (−0.5 − 0.866i)8-s + (−0.347 + 1.96i)9-s + (−1.5 − 2.59i)11-s + (0.499 − 0.866i)12-s + (−1.53 − 1.28i)13-s + (3.75 + 1.36i)14-s + (0.766 − 0.642i)16-s + (−1.04 − 5.90i)17-s − 2·18-s + (0.694 + 3.93i)21-s + (2.29 − 1.92i)22-s + ⋯
L(s)  = 1  + (0.122 + 0.696i)2-s + (−0.442 + 0.371i)3-s + (−0.469 + 0.171i)4-s + (−0.312 − 0.262i)6-s + (0.755 − 1.30i)7-s + (−0.176 − 0.306i)8-s + (−0.115 + 0.656i)9-s + (−0.452 − 0.783i)11-s + (0.144 − 0.249i)12-s + (−0.424 − 0.356i)13-s + (1.00 + 0.365i)14-s + (0.191 − 0.160i)16-s + (−0.252 − 1.43i)17-s − 0.471·18-s + (0.151 + 0.859i)21-s + (0.489 − 0.411i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.910 + 0.413i$
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: \((2,\ 722,\ (\ :1/2),\ 0.910 + 0.413i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08463 - 0.234790i\)
\(L(\frac12)\) \(\approx\) \(1.08463 - 0.234790i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.766 - 0.642i)T + (0.520 - 2.95i)T^{2} \)
5 \( 1 + (3.83 + 3.21i)T^{2} \)
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 \( 1 + (1.53 + 1.28i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.04 + 5.90i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-5.63 + 2.05i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + (6.89 - 5.78i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-3.75 - 1.36i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-5.63 + 2.05i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-1.56 - 8.86i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-3.75 + 1.36i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-1.21 + 6.89i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (5.63 + 2.05i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (0.766 - 0.642i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-3.06 + 2.57i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (1.5 - 2.59i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.59 + 3.85i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (2.95 + 16.7i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.41151118760917549718589166798, −9.578653415585766164435895433241, −8.313211767042462124127398063282, −7.71487877702511920254733829794, −6.95485436276441451289361155692, −5.74411505271137136788480307135, −4.87377015265457547743111652272, −4.32101285497811208830324303375, −2.78987004514008445659399093857, −0.60927823552172014756177828807, 1.55747643481342866717932167875, 2.52679004143879637951615588848, 3.94931704941468278148591978383, 5.13451626710826293614325397983, 5.77191525027095326216726063573, 6.87931418683596034150518628212, 8.012776634527695323240246324896, 8.925080737640014908189936141855, 9.577028814983805895844011040628, 10.66686588995382077728908510514

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