Properties

Label 2-684-171.11-c2-0-32
Degree $2$
Conductor $684$
Sign $-0.433 + 0.901i$
Analytic cond. $18.6376$
Root an. cond. $4.31713$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.79 + 2.40i)3-s − 4.55i·5-s + (5.85 − 10.1i)7-s + (−2.55 − 8.63i)9-s + (14.0 + 8.12i)11-s + (−3.63 + 6.29i)13-s + (10.9 + 8.17i)15-s + (−19.6 − 11.3i)17-s + (−13.5 − 13.2i)19-s + (13.8 + 32.2i)21-s + (−33.4 − 19.3i)23-s + 4.28·25-s + (25.3 + 9.37i)27-s + 25.8i·29-s + (14.0 + 24.3i)31-s + ⋯
L(s)  = 1  + (−0.598 + 0.801i)3-s − 0.910i·5-s + (0.835 − 1.44i)7-s + (−0.283 − 0.959i)9-s + (1.27 + 0.738i)11-s + (−0.279 + 0.483i)13-s + (0.729 + 0.544i)15-s + (−1.15 − 0.666i)17-s + (−0.715 − 0.698i)19-s + (0.659 + 1.53i)21-s + (−1.45 − 0.840i)23-s + 0.171·25-s + (0.937 + 0.347i)27-s + 0.890i·29-s + (0.452 + 0.784i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.433 + 0.901i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ -0.433 + 0.901i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.015120456\)
\(L(\frac12)\) \(\approx\) \(1.015120456\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.79 - 2.40i)T \)
19 \( 1 + (13.5 + 13.2i)T \)
good5 \( 1 + 4.55iT - 25T^{2} \)
7 \( 1 + (-5.85 + 10.1i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-14.0 - 8.12i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (3.63 - 6.29i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (19.6 + 11.3i)T + (144.5 + 250. i)T^{2} \)
23 \( 1 + (33.4 + 19.3i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 25.8iT - 841T^{2} \)
31 \( 1 + (-14.0 - 24.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 47.8T + 1.36e3T^{2} \)
41 \( 1 + 8.35iT - 1.68e3T^{2} \)
43 \( 1 + (39.1 + 67.8i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 - 16.4iT - 2.20e3T^{2} \)
53 \( 1 + (-43.6 + 25.1i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 - 74.9iT - 3.48e3T^{2} \)
61 \( 1 - 62.3T + 3.72e3T^{2} \)
67 \( 1 + (-28.5 + 49.4i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (73.0 + 42.1i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-28.4 + 49.2i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (59.5 + 103. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (71.4 + 41.2i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-123. + 71.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (59.2 + 102. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.21511619351761663319247013291, −9.027888312019707985696263636101, −8.651983069482349790035965952481, −7.07819346086720342596239946472, −6.62533459664506955656980235791, −4.98656215732478572574126024417, −4.51119891780744578604603650535, −3.91728381812094395178642772001, −1.71076779175834766134551103170, −0.38962953943912822250359540170, 1.67522423230247109388207556543, 2.53201080305898024838219689804, 4.06316349982794795741446902280, 5.49453294799601316217037148480, 6.14964857215750449416364873895, 6.77162975785056269014047572886, 8.163211639740591441414606201795, 8.434786722274866483325610805944, 9.772864238334421732807849624347, 10.85836501630578031039337388029

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