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Properties

Label 2-722-19.7-c1-0-26
Degree $2$
Conductor $722$
Sign $0.910 - 0.412i$
Analytic cond. $5.76519$
Root an. cond. $2.40108$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−1.5 − 2.59i)3-s + (−0.499 + 0.866i)4-s + (−1 − 1.73i)5-s + (−1.5 + 2.59i)6-s − 3·7-s + 0.999·8-s + (−3 + 5.19i)9-s + (−0.999 + 1.73i)10-s − 2·11-s + 3·12-s + (1.5 − 2.59i)13-s + (1.5 + 2.59i)14-s + (−3 + 5.19i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.866 − 1.49i)3-s + (−0.249 + 0.433i)4-s + (−0.447 − 0.774i)5-s + (−0.612 + 1.06i)6-s − 1.13·7-s + 0.353·8-s + (−1 + 1.73i)9-s + (−0.316 + 0.547i)10-s − 0.603·11-s + 0.866·12-s + (0.416 − 0.720i)13-s + (0.400 + 0.694i)14-s + (−0.774 + 1.34i)15-s + (−0.125 − 0.216i)16-s + (0.121 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.910 - 0.412i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (653, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ 0.910 - 0.412i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 \)
good3 \( 1 + (1.5 + 2.59i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 3T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + (-1.5 + 2.59i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.5 - 4.33i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + (6 + 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.5 - 12.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6 - 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6 + 10.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.722554163870129623924253296631, −8.567713903351625474916022369204, −7.86797887636509858048579923304, −7.09992574308365863368708802556, −6.03608112816006667343922363341, −5.30938132427389845527658135763, −3.77767311375806387082244958360, −2.44429873639467994954685809362, −1.02823271021404005042513044812, 0, 3.06142050202212257141506903621, 3.99209228214361539135954651834, 4.95124356927901601576876201963, 6.04372917736502442149991096708, 6.56714984365498803193934831293, 7.63670174069461568158016305560, 8.980756612361271874490696018293, 9.518580032017582096129309699952, 10.44352285086300015420734139080

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