Properties

Label 2-684-4.3-c2-0-41
Degree $2$
Conductor $684$
Sign $0.971 + 0.238i$
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  + (−1.57 + 1.23i)2-s + (0.954 − 3.88i)4-s − 3.66·5-s + 1.93i·7-s + (3.29 + 7.29i)8-s + (5.76 − 4.51i)10-s − 0.752i·11-s − 13.0·13-s + (−2.38 − 3.04i)14-s + (−14.1 − 7.41i)16-s + 23.9·17-s − 4.35i·19-s + (−3.49 + 14.2i)20-s + (0.928 + 1.18i)22-s + 6.26i·23-s + ⋯
L(s)  = 1  + (−0.786 + 0.616i)2-s + (0.238 − 0.971i)4-s − 0.732·5-s + 0.276i·7-s + (0.411 + 0.911i)8-s + (0.576 − 0.451i)10-s − 0.0684i·11-s − 1.00·13-s + (−0.170 − 0.217i)14-s + (−0.886 − 0.463i)16-s + 1.40·17-s − 0.229i·19-s + (−0.174 + 0.711i)20-s + (0.0422 + 0.0538i)22-s + 0.272i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.238i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.971 + 0.238i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ 0.971 + 0.238i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8082492921\)
\(L(\frac12)\) \(\approx\) \(0.8082492921\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.57 - 1.23i)T \)
3 \( 1 \)
19 \( 1 + 4.35iT \)
good5 \( 1 + 3.66T + 25T^{2} \)
7 \( 1 - 1.93iT - 49T^{2} \)
11 \( 1 + 0.752iT - 121T^{2} \)
13 \( 1 + 13.0T + 169T^{2} \)
17 \( 1 - 23.9T + 289T^{2} \)
23 \( 1 - 6.26iT - 529T^{2} \)
29 \( 1 + 33.1T + 841T^{2} \)
31 \( 1 - 17.5iT - 961T^{2} \)
37 \( 1 - 41.5T + 1.36e3T^{2} \)
41 \( 1 - 5.51T + 1.68e3T^{2} \)
43 \( 1 + 84.5iT - 1.84e3T^{2} \)
47 \( 1 + 18.3iT - 2.20e3T^{2} \)
53 \( 1 - 41.2T + 2.80e3T^{2} \)
59 \( 1 + 69.5iT - 3.48e3T^{2} \)
61 \( 1 - 87.6T + 3.72e3T^{2} \)
67 \( 1 + 105. iT - 4.48e3T^{2} \)
71 \( 1 + 74.9iT - 5.04e3T^{2} \)
73 \( 1 - 48.7T + 5.32e3T^{2} \)
79 \( 1 - 95.9iT - 6.24e3T^{2} \)
83 \( 1 - 65.8iT - 6.88e3T^{2} \)
89 \( 1 + 3.38T + 7.92e3T^{2} \)
97 \( 1 + 23.5T + 9.40e3T^{2} \)
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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

−10.00291221917951418279987664073, −9.385627428678342047824825526675, −8.376187198819928757004768598951, −7.62455567246623802501858287252, −7.05650806618666644249439179623, −5.77886669027525812573314941350, −5.05922588888340521775601744188, −3.66767020194969754007063190800, −2.16095329940405999303563058815, −0.51411816592709628896973315380, 0.895753647867246518157378979753, 2.43614286409568021185739503722, 3.58323736176239306030240292873, 4.46041295761632494531034174192, 5.90312240589764286756673405910, 7.43938609415844417313110959526, 7.56238045529165577365191507484, 8.593175350321748661681375708770, 9.726322151428736008780901968505, 10.08314108395384425372294029436

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