Properties

Label 2-684-228.227-c2-0-56
Degree $2$
Conductor $684$
Sign $-0.451 + 0.892i$
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  + (−1.90 − 0.613i)2-s + (3.24 + 2.33i)4-s − 6.64i·5-s + 0.470i·7-s + (−4.74 − 6.43i)8-s + (−4.07 + 12.6i)10-s + 10.1·11-s + 3.45i·13-s + (0.288 − 0.896i)14-s + (5.09 + 15.1i)16-s − 14.2i·17-s + (13.6 − 13.2i)19-s + (15.5 − 21.5i)20-s + (−19.3 − 6.24i)22-s + 25.5·23-s + ⋯
L(s)  = 1  + (−0.951 − 0.306i)2-s + (0.811 + 0.583i)4-s − 1.32i·5-s + 0.0672i·7-s + (−0.593 − 0.804i)8-s + (−0.407 + 1.26i)10-s + 0.924·11-s + 0.265i·13-s + (0.0206 − 0.0640i)14-s + (0.318 + 0.948i)16-s − 0.839i·17-s + (0.717 − 0.696i)19-s + (0.775 − 1.07i)20-s + (−0.880 − 0.283i)22-s + 1.11·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.451 + 0.892i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ -0.451 + 0.892i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.116764317\)
\(L(\frac12)\) \(\approx\) \(1.116764317\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.90 + 0.613i)T \)
3 \( 1 \)
19 \( 1 + (-13.6 + 13.2i)T \)
good5 \( 1 + 6.64iT - 25T^{2} \)
7 \( 1 - 0.470iT - 49T^{2} \)
11 \( 1 - 10.1T + 121T^{2} \)
13 \( 1 - 3.45iT - 169T^{2} \)
17 \( 1 + 14.2iT - 289T^{2} \)
23 \( 1 - 25.5T + 529T^{2} \)
29 \( 1 + 14.1T + 841T^{2} \)
31 \( 1 + 19.3T + 961T^{2} \)
37 \( 1 - 30.0iT - 1.36e3T^{2} \)
41 \( 1 - 25.6T + 1.68e3T^{2} \)
43 \( 1 + 38.5iT - 1.84e3T^{2} \)
47 \( 1 + 16.5T + 2.20e3T^{2} \)
53 \( 1 + 41.6T + 2.80e3T^{2} \)
59 \( 1 + 31.3iT - 3.48e3T^{2} \)
61 \( 1 + 3.53T + 3.72e3T^{2} \)
67 \( 1 + 13.9T + 4.48e3T^{2} \)
71 \( 1 + 128. iT - 5.04e3T^{2} \)
73 \( 1 + 71.6T + 5.32e3T^{2} \)
79 \( 1 - 100.T + 6.24e3T^{2} \)
83 \( 1 + 58.6T + 6.88e3T^{2} \)
89 \( 1 + 108.T + 7.92e3T^{2} \)
97 \( 1 + 124. iT - 9.40e3T^{2} \)
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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

−9.580304556943486597792129375317, −9.199492455699836322472661300162, −8.604446886845650138216962193222, −7.50637562389324875976308845610, −6.73270542569531295491930075513, −5.43693956212027776238495505328, −4.39950577848334976174195016658, −3.10851016021292820318179746210, −1.60708251336224015375230387784, −0.61023278897012845997477440787, 1.34642534650225139188619731860, 2.71698835744317045121585703467, 3.77854844465640577074856231188, 5.55265074284091488433506239650, 6.37968697523063449759909981196, 7.13388828112565758546114730882, 7.81131094929506821871461066777, 8.925027008902362607969327175059, 9.677649201574949724599268322421, 10.56163204504613876261771398706

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