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Properties

Label 2-722-19.6-c1-0-8
Degree $2$
Conductor $722$
Sign $-0.412 - 0.910i$
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  + (−0.766 + 0.642i)2-s + (1.20 + 0.439i)3-s + (0.173 − 0.984i)4-s + (0.641 + 3.63i)5-s + (−1.20 + 0.439i)6-s + (0.221 − 0.383i)7-s + (0.500 + 0.866i)8-s + (−1.03 − 0.868i)9-s + (−2.83 − 2.37i)10-s + (2.01 + 3.48i)11-s + (0.642 − 1.11i)12-s + (4.59 − 1.67i)13-s + (0.0768 + 0.435i)14-s + (−0.824 + 4.67i)15-s + (−0.939 − 0.342i)16-s + (0.204 − 0.171i)17-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (0.696 + 0.253i)3-s + (0.0868 − 0.492i)4-s + (0.287 + 1.62i)5-s + (−0.492 + 0.179i)6-s + (0.0836 − 0.144i)7-s + (0.176 + 0.306i)8-s + (−0.345 − 0.289i)9-s + (−0.895 − 0.751i)10-s + (0.607 + 1.05i)11-s + (0.185 − 0.321i)12-s + (1.27 − 0.464i)13-s + (0.0205 + 0.116i)14-s + (−0.212 + 1.20i)15-s + (−0.234 − 0.0855i)16-s + (0.0495 − 0.0416i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $-0.412 - 0.910i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (595, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ -0.412 - 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.828785 + 1.28541i\)
\(L(\frac12)\) \(\approx\) \(0.828785 + 1.28541i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 \)
good3 \( 1 + (-1.20 - 0.439i)T + (2.29 + 1.92i)T^{2} \)
5 \( 1 + (-0.641 - 3.63i)T + (-4.69 + 1.71i)T^{2} \)
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 \( 1 + (-4.59 + 1.67i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-0.204 + 0.171i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (1.59 - 9.06i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-0.171 - 0.143i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-1.73 + 3.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.44T + 37T^{2} \)
41 \( 1 + (7.39 + 2.68i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.978 + 5.54i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-1.68 - 1.41i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (1.72 - 9.79i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-2.68 + 2.25i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-0.708 + 4.01i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (0.113 + 0.0949i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.99 + 11.2i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (1.33 + 0.486i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-9.58 - 3.48i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-1.64 + 2.84i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.60 - 2.03i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (3.12 - 2.62i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.47675950587783664658653480062, −9.717178480122506359675573032216, −9.092768146388730555509671977540, −7.999791548602563673573372824555, −7.26817530202576514137652745739, −6.43402274579687189358567986890, −5.68001307338622193173503795964, −3.91439798168054838835381105791, −3.12868678221070411342663969462, −1.83211250542855158448180827818, 0.934199539526574832160565747834, 1.99256572212275942211374448471, 3.37887009829581101670166761947, 4.47298719799706378045317386260, 5.64002013975080306652370339894, 6.64684736147663940122375156463, 8.297382402993849715542764164070, 8.496911794573822705009495959094, 8.882699868886773680772296516719, 9.938148736282664661767610879972

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