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Properties

Label 2-722-19.16-c1-0-3
Degree $2$
Conductor $722$
Sign $0.745 - 0.666i$
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 + (0.415 − 0.151i)3-s + (0.173 + 0.984i)4-s + (−0.154 + 0.877i)5-s + (−0.415 − 0.151i)6-s + (−1.26 − 2.18i)7-s + (0.500 − 0.866i)8-s + (−2.14 + 1.80i)9-s + (0.682 − 0.573i)10-s + (−0.975 + 1.69i)11-s + (0.221 + 0.383i)12-s + (6.07 + 2.20i)13-s + (−0.437 + 2.48i)14-s + (0.0684 + 0.388i)15-s + (−0.939 + 0.342i)16-s + (−2.62 − 2.19i)17-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (0.240 − 0.0873i)3-s + (0.0868 + 0.492i)4-s + (−0.0692 + 0.392i)5-s + (−0.169 − 0.0617i)6-s + (−0.476 − 0.824i)7-s + (0.176 − 0.306i)8-s + (−0.716 + 0.600i)9-s + (0.215 − 0.181i)10-s + (−0.294 + 0.509i)11-s + (0.0638 + 0.110i)12-s + (1.68 + 0.612i)13-s + (−0.116 + 0.663i)14-s + (0.0176 + 0.100i)15-s + (−0.234 + 0.0855i)16-s + (−0.635 − 0.533i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.745 - 0.666i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ 0.745 - 0.666i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.932371 + 0.356324i\)
\(L(\frac12)\) \(\approx\) \(0.932371 + 0.356324i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.415 + 0.151i)T + (2.29 - 1.92i)T^{2} \)
5 \( 1 + (0.154 - 0.877i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (1.26 + 2.18i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.975 - 1.69i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-6.07 - 2.20i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (2.62 + 2.19i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-1.42 - 8.06i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-3.50 + 2.94i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-4.39 - 7.61i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.97T + 37T^{2} \)
41 \( 1 + (-3.27 + 1.19i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.08 - 6.15i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (8.08 - 6.78i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-0.653 - 3.70i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-2.17 - 1.82i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (0.426 + 2.41i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (2.04 - 1.71i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.00980 + 0.0555i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-6.54 + 2.38i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-8.58 + 3.12i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-7.06 - 12.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.75 + 0.638i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-5.99 - 5.02i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.68957382416017083486184558192, −9.645810410192209589618072626548, −8.852659806583506605516693295651, −8.032126027545245867159967399798, −7.10841865753592276849181757955, −6.38670464385663489868185796330, −4.93402029861118211924556329758, −3.67792024691093586570754860466, −2.85684103178950495239119065499, −1.39998476687121448813776878238, 0.65023475810920884657992159237, 2.54921410486005117310603324486, 3.64713079526849731528629430831, 5.09865125627541085198843223543, 6.18975615192019270243419040870, 6.45680609328631944648506688695, 8.190365878097600301708065450817, 8.634733805637900194209792403415, 8.998610665153482324023097779064, 10.27229121272058699689867996142

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