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Properties

Label 2-684-228.11-c1-0-8
Degree $2$
Conductor $684$
Sign $-0.770 - 0.637i$
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  + (−0.457 + 1.33i)2-s + (−1.58 − 1.22i)4-s + (3.60 + 2.08i)5-s + 4.46i·7-s + (2.36 − 1.55i)8-s + (−4.43 + 3.87i)10-s − 1.16·11-s + (0.545 + 0.945i)13-s + (−5.97 − 2.04i)14-s + (0.997 + 3.87i)16-s + (−1.65 − 0.953i)17-s + (2.72 − 3.39i)19-s + (−3.15 − 7.71i)20-s + (0.534 − 1.56i)22-s + (−1.17 − 2.02i)23-s + ⋯
L(s)  = 1  + (−0.323 + 0.946i)2-s + (−0.790 − 0.612i)4-s + (1.61 + 0.931i)5-s + 1.68i·7-s + (0.835 − 0.549i)8-s + (−1.40 + 1.22i)10-s − 0.351·11-s + (0.151 + 0.262i)13-s + (−1.59 − 0.546i)14-s + (0.249 + 0.968i)16-s + (−0.400 − 0.231i)17-s + (0.626 − 0.779i)19-s + (−0.704 − 1.72i)20-s + (0.113 − 0.332i)22-s + (−0.244 − 0.422i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.770 - 0.637i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ -0.770 - 0.637i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.506465 + 1.40663i\)
\(L(\frac12)\) \(\approx\) \(0.506465 + 1.40663i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.457 - 1.33i)T \)
3 \( 1 \)
19 \( 1 + (-2.72 + 3.39i)T \)
good5 \( 1 + (-3.60 - 2.08i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 4.46iT - 7T^{2} \)
11 \( 1 + 1.16T + 11T^{2} \)
13 \( 1 + (-0.545 - 0.945i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.65 + 0.953i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.17 + 2.02i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.74 + 4.47i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.08iT - 31T^{2} \)
37 \( 1 + 6.77T + 37T^{2} \)
41 \( 1 + (4.90 + 2.83i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.18 + 3.57i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.985 - 1.70i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.680 - 0.393i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.09 + 8.82i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.42 - 9.39i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.26 - 0.732i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.75 + 3.03i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.29 - 9.17i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-10.2 - 5.92i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.42T + 83T^{2} \)
89 \( 1 + (-5.94 + 3.43i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.490 - 0.848i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.39365386595588056282661920959, −9.848516916915902683958054502932, −8.975131706903633451768101670134, −8.478351883354667368361824097241, −6.94464965793877856446675410346, −6.45839802564114681137375810472, −5.56071018710779353060125890315, −5.02550179983980914958883928806, −2.91627561731487259240162599516, −1.94881291219779695657386239024, 0.945002975095355696839734836223, 1.86332999299579929590352825764, 3.37613477409652219814834381828, 4.54461086119325657898198525535, 5.33331555496338581090358863360, 6.59239613633046967394456980854, 7.81323156380062425868057562221, 8.619661959817715511889598803978, 9.607571222828710267216554659296, 10.21685771864343325956819087763

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