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Properties

Label 2-684-171.49-c1-0-10
Degree $2$
Conductor $684$
Sign $0.694 + 0.719i$
Analytic cond. $5.46176$
Root an. cond. $2.33704$
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  + (−0.833 + 1.51i)3-s + (−1.50 − 2.60i)5-s + (0.886 + 1.53i)7-s + (−1.61 − 2.53i)9-s + (1.38 + 2.39i)11-s − 2.80·13-s + (5.20 − 0.113i)15-s + (3.80 − 6.58i)17-s + (−3.30 − 2.84i)19-s + (−3.06 + 0.0668i)21-s + 4.08·23-s + (−2.01 + 3.49i)25-s + (5.18 − 0.339i)27-s + (4.08 − 7.07i)29-s + (1.70 − 2.95i)31-s + ⋯
L(s)  = 1  + (−0.481 + 0.876i)3-s + (−0.672 − 1.16i)5-s + (0.334 + 0.580i)7-s + (−0.537 − 0.843i)9-s + (0.416 + 0.721i)11-s − 0.777·13-s + (1.34 − 0.0292i)15-s + (0.922 − 1.59i)17-s + (−0.758 − 0.651i)19-s + (−0.669 + 0.0145i)21-s + 0.851·23-s + (−0.403 + 0.699i)25-s + (0.997 − 0.0652i)27-s + (0.758 − 1.31i)29-s + (0.306 − 0.530i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 + 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.694 + 0.719i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ 0.694 + 0.719i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.898426 - 0.381167i\)
\(L(\frac12)\) \(\approx\) \(0.898426 - 0.381167i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.833 - 1.51i)T \)
19 \( 1 + (3.30 + 2.84i)T \)
good5 \( 1 + (1.50 + 2.60i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.886 - 1.53i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.38 - 2.39i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.80T + 13T^{2} \)
17 \( 1 + (-3.80 + 6.58i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 - 4.08T + 23T^{2} \)
29 \( 1 + (-4.08 + 7.07i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.70 + 2.95i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.18T + 37T^{2} \)
41 \( 1 + (1.12 + 1.95i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 4.62T + 43T^{2} \)
47 \( 1 + (-2.06 + 3.57i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.59 - 6.23i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.714 - 1.23i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.49 - 7.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 + (-1.75 + 3.04i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.37 + 11.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 8.04T + 79T^{2} \)
83 \( 1 + (6.03 + 10.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.76 - 3.05i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.6T + 97T^{2} \)
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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

−10.23020106835738564329307450385, −9.427573949703482491890478931292, −8.883995132992730394138165878983, −7.900976352537707049270722650244, −6.83301269637205276796496803994, −5.47130886942182357070717922102, −4.80069116846679321923148408232, −4.23394671869399690510634937516, −2.68081000000962312741337415510, −0.61888334687639562994026580970, 1.32718326401342096279626399286, 2.90680936492367000327244717818, 3.93011887361924838837186107777, 5.31290343222703585574246822104, 6.46052527478157543454678878853, 6.94741934449062321334946774615, 7.912949718855220826590840176456, 8.436894999655551142903872343466, 10.10906050448585773313328172817, 10.82425685537549950527416889657

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