Properties

Label 2-722-19.11-c1-0-15
Degree $2$
Conductor $722$
Sign $0.519 + 0.854i$
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 + (0.221 − 0.383i)3-s + (−0.499 − 0.866i)4-s + (0.445 − 0.772i)5-s + (−0.221 − 0.383i)6-s + 2.52·7-s − 0.999·8-s + (1.40 + 2.42i)9-s + (−0.445 − 0.772i)10-s + 1.95·11-s − 0.442·12-s + (3.22 + 5.59i)13-s + (1.26 − 2.18i)14-s + (−0.197 − 0.341i)15-s + (−0.5 + 0.866i)16-s + (1.71 − 2.96i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.127 − 0.221i)3-s + (−0.249 − 0.433i)4-s + (0.199 − 0.345i)5-s + (−0.0903 − 0.156i)6-s + 0.952·7-s − 0.353·8-s + (0.467 + 0.809i)9-s + (−0.140 − 0.244i)10-s + 0.588·11-s − 0.127·12-s + (0.895 + 1.55i)13-s + (0.336 − 0.583i)14-s + (−0.0509 − 0.0882i)15-s + (−0.125 + 0.216i)16-s + (0.415 − 0.718i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.519 + 0.854i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (429, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ 0.519 + 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97688 - 1.11180i\)
\(L(\frac12)\) \(\approx\) \(1.97688 - 1.11180i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.221 + 0.383i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.445 + 0.772i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 2.52T + 7T^{2} \)
11 \( 1 - 1.95T + 11T^{2} \)
13 \( 1 + (-3.22 - 5.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.71 + 2.96i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (4.09 + 7.08i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.29 + 3.96i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.79T + 31T^{2} \)
37 \( 1 + 5.97T + 37T^{2} \)
41 \( 1 + (-1.74 + 3.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.12 + 5.40i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.27 - 9.13i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.88 + 3.25i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.42 - 2.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.22 - 2.12i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.33 - 2.31i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.0282 - 0.0488i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.48 + 6.03i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.56 + 7.90i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.1T + 83T^{2} \)
89 \( 1 + (0.933 + 1.61i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.91 - 6.77i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.51693806591608014183675090467, −9.302939119682947097492505974069, −8.783090833533138817219764517988, −7.69987067531844872004608057808, −6.73838607886392925784348965324, −5.56913010685035324371332483587, −4.60894489128934444182065278815, −3.90520899979445098405627888533, −2.17090470355986532715872659854, −1.43802281331933286445902747949, 1.45770859958325146078106448008, 3.35762292552573694488109964729, 3.97895459892587577048609887508, 5.34560348001099619698908866577, 5.98038209080096317612001323226, 7.03919039252889841292563623934, 7.933032026531586279916530607560, 8.647418410542743251988503106783, 9.629107775263755609357147235888, 10.55361586617873425003509936342

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