Properties

Label 2-722-19.16-c1-0-14
Degree $2$
Conductor $722$
Sign $0.713 + 0.700i$
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 + (−0.939 + 0.342i)3-s + (0.173 + 0.984i)4-s + (0.939 + 0.342i)6-s + (0.5 + 0.866i)7-s + (0.500 − 0.866i)8-s + (−1.53 + 1.28i)9-s + (3 − 5.19i)11-s + (−0.499 − 0.866i)12-s + (−4.69 − 1.71i)13-s + (0.173 − 0.984i)14-s + (−0.939 + 0.342i)16-s + (2.29 + 1.92i)17-s + 2·18-s + (−0.766 − 0.642i)21-s + (−5.63 + 2.05i)22-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (−0.542 + 0.197i)3-s + (0.0868 + 0.492i)4-s + (0.383 + 0.139i)6-s + (0.188 + 0.327i)7-s + (0.176 − 0.306i)8-s + (−0.510 + 0.428i)9-s + (0.904 − 1.56i)11-s + (−0.144 − 0.250i)12-s + (−1.30 − 0.474i)13-s + (0.0464 − 0.263i)14-s + (−0.234 + 0.0855i)16-s + (0.557 + 0.467i)17-s + 0.471·18-s + (−0.167 − 0.140i)21-s + (−1.20 + 0.437i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.713 + 0.700i$
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: \((2,\ 722,\ (\ :1/2),\ 0.713 + 0.700i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.823912 - 0.336786i\)
\(L(\frac12)\) \(\approx\) \(0.823912 - 0.336786i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.939 - 0.342i)T + (2.29 - 1.92i)T^{2} \)
5 \( 1 + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.69 + 1.71i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-2.29 - 1.92i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-0.520 - 2.95i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-6.89 + 5.78i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.38 + 7.87i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (0.520 + 2.95i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-6.89 - 5.78i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.73 + 9.84i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-3.83 + 3.21i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.04 - 5.90i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-6.57 + 2.39i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-9.39 + 3.42i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-3 - 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-11.2 - 4.10i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (7.66 + 6.42i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.42764812410789024455652544666, −9.537742758530019945787524772543, −8.553823103621976347840663416240, −8.040047921444580013609331157888, −6.77250980085303357205236841764, −5.76256742501649678668128197245, −4.95952966688914636675416574951, −3.54176391018470446930257296287, −2.51046720756697611570296529813, −0.76654460221100648959069070568, 1.05798373296979343332913414203, 2.60994540278981947906347185091, 4.42820155366933605898206969031, 5.08690080424406260279723076106, 6.46116076186513131800371845171, 6.91448431750206548763216769880, 7.73229473201446854601812596115, 8.922924186678944890479747329409, 9.605264032982960762349398425599, 10.32471671956361464494070609262

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