Properties

Label 2-684-4.3-c2-0-24
Degree $2$
Conductor $684$
Sign $-0.156 - 0.987i$
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  + (0.156 + 1.99i)2-s + (−3.95 + 0.624i)4-s − 5.18·5-s − 9.10i·7-s + (−1.86 − 7.77i)8-s + (−0.812 − 10.3i)10-s + 9.19i·11-s − 3.83·13-s + (18.1 − 1.42i)14-s + (15.2 − 4.93i)16-s + 7.85·17-s − 4.35i·19-s + (20.4 − 3.24i)20-s + (−18.3 + 1.44i)22-s + 8.61i·23-s + ⋯
L(s)  = 1  + (0.0783 + 0.996i)2-s + (−0.987 + 0.156i)4-s − 1.03·5-s − 1.30i·7-s + (−0.233 − 0.972i)8-s + (−0.0812 − 1.03i)10-s + 0.835i·11-s − 0.294·13-s + (1.29 − 0.101i)14-s + (0.951 − 0.308i)16-s + 0.461·17-s − 0.229i·19-s + (1.02 − 0.162i)20-s + (−0.833 + 0.0654i)22-s + 0.374i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.156 - 0.987i$
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: \((2,\ 684,\ (\ :1),\ -0.156 - 0.987i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.096058966\)
\(L(\frac12)\) \(\approx\) \(1.096058966\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.156 - 1.99i)T \)
3 \( 1 \)
19 \( 1 + 4.35iT \)
good5 \( 1 + 5.18T + 25T^{2} \)
7 \( 1 + 9.10iT - 49T^{2} \)
11 \( 1 - 9.19iT - 121T^{2} \)
13 \( 1 + 3.83T + 169T^{2} \)
17 \( 1 - 7.85T + 289T^{2} \)
23 \( 1 - 8.61iT - 529T^{2} \)
29 \( 1 - 30.9T + 841T^{2} \)
31 \( 1 - 43.4iT - 961T^{2} \)
37 \( 1 - 59.9T + 1.36e3T^{2} \)
41 \( 1 + 1.02T + 1.68e3T^{2} \)
43 \( 1 - 7.62iT - 1.84e3T^{2} \)
47 \( 1 - 45.5iT - 2.20e3T^{2} \)
53 \( 1 + 65.9T + 2.80e3T^{2} \)
59 \( 1 - 59.2iT - 3.48e3T^{2} \)
61 \( 1 - 67.6T + 3.72e3T^{2} \)
67 \( 1 + 81.0iT - 4.48e3T^{2} \)
71 \( 1 - 109. iT - 5.04e3T^{2} \)
73 \( 1 - 49.8T + 5.32e3T^{2} \)
79 \( 1 + 2.49iT - 6.24e3T^{2} \)
83 \( 1 + 134. iT - 6.88e3T^{2} \)
89 \( 1 + 67.1T + 7.92e3T^{2} \)
97 \( 1 - 116.T + 9.40e3T^{2} \)
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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

−10.29651193340992743425423243356, −9.642989694886775246642463406989, −8.472253605086614995881775788184, −7.58173894587713623339410437718, −7.28332585071028088385775768668, −6.29107295897963740319991164788, −4.84015095539925051256316608303, −4.29473747057315228918626035278, −3.30540185212847036165099740409, −0.897227239840434587163424530142, 0.55062564974443163481647408537, 2.30465971375416846308685144687, 3.25088977163712182841422597749, 4.25577389562324226504217899287, 5.33001935302425679202806169173, 6.21806809529782429532869774401, 7.88814738644772554617062321447, 8.355248089087107779325023287409, 9.259677010354234266574028698487, 10.08114106760071396620499876634

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