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Properties

Label 2-722-19.7-c1-0-7
Degree $2$
Conductor $722$
Sign $0.813 + 0.582i$
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  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.499 − 0.866i)6-s − 4·7-s + 0.999·8-s + (1 − 1.73i)9-s + 3·11-s − 0.999·12-s + (1 − 1.73i)13-s + (2 + 3.46i)14-s + (−0.5 − 0.866i)16-s + (3 + 5.19i)17-s − 2·18-s + (−2 − 3.46i)21-s + (−1.5 − 2.59i)22-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.204 − 0.353i)6-s − 1.51·7-s + 0.353·8-s + (0.333 − 0.577i)9-s + 0.904·11-s − 0.288·12-s + (0.277 − 0.480i)13-s + (0.534 + 0.925i)14-s + (−0.125 − 0.216i)16-s + (0.727 + 1.26i)17-s − 0.471·18-s + (−0.436 − 0.755i)21-s + (−0.319 − 0.553i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.813 + 0.582i$
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: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ 0.813 + 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21118 - 0.388971i\)
\(L(\frac12)\) \(\approx\) \(1.21118 - 0.388971i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 4T + 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.5 - 14.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.15761349996729779102090249759, −9.568211410120342513242644577001, −8.934927884322279446399086143565, −8.005691707083077555964883468116, −6.63058751192366131405794879734, −6.14010503213585919367476995879, −4.39614871962962577023010062234, −3.60308736951225359916574630362, −2.83705727483024362035365225448, −0.944022741958428169397854814197, 1.14546742610286588241836353242, 2.79886206316725918712261368610, 3.96951993316930154057935375907, 5.34119516713738043799628797589, 6.30792894084654795247437044598, 7.18781469341429282191450835697, 7.52892217139477526881306715143, 9.094741872857401152364938958964, 9.247545614676605058209101754583, 10.22062006650143921146447576915

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