Properties

Label 2-684-19.12-c2-0-10
Degree $2$
Conductor $684$
Sign $0.131 + 0.991i$
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  − 11·7-s + (22.5 + 12.9i)13-s + (−13 − 13.8i)19-s + (12.5 − 21.6i)25-s − 60.6i·31-s − 57.1i·37-s + (−41.5 − 71.8i)43-s + 72·49-s + (−60.5 + 104. i)61-s + (115.5 + 66.6i)67-s + (−71.5 − 123. i)73-s + (76.5 − 44.1i)79-s + (−247.5 − 142. i)91-s + (168 − 96.9i)97-s − 133. i·103-s + ⋯
L(s)  = 1  − 1.57·7-s + (1.73 + 0.999i)13-s + (−0.684 − 0.729i)19-s + (0.5 − 0.866i)25-s − 1.95i·31-s − 1.54i·37-s + (−0.965 − 1.67i)43-s + 1.46·49-s + (−0.991 + 1.71i)61-s + (1.72 + 0.995i)67-s + (−0.979 − 1.69i)73-s + (0.968 − 0.559i)79-s + (−2.71 − 1.57i)91-s + (1.73 − 0.999i)97-s − 1.29i·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.131 + 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.131 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.131 + 0.991i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ 0.131 + 0.991i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.125175280\)
\(L(\frac12)\) \(\approx\) \(1.125175280\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + (13 + 13.8i)T \)
good5 \( 1 + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + 11T + 49T^{2} \)
11 \( 1 + 121T^{2} \)
13 \( 1 + (-22.5 - 12.9i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (420.5 + 728. i)T^{2} \)
31 \( 1 + 60.6iT - 961T^{2} \)
37 \( 1 + 57.1iT - 1.36e3T^{2} \)
41 \( 1 + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (41.5 + 71.8i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (60.5 - 104. i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-115.5 - 66.6i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (71.5 + 123. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-76.5 + 44.1i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 6.88e3T^{2} \)
89 \( 1 + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-168 + 96.9i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.07134004508359129576049360505, −9.094984965812344253487554075684, −8.666043772186825028148405398702, −7.25627531938221094505396265960, −6.40952927209120542109622744322, −5.90565837167600327482695798339, −4.27690279885097168009028848205, −3.53294852899361318296913311927, −2.24360501495517763523392124814, −0.44371192407274583291046544366, 1.22947968115215383432302855546, 3.13300107982041113196853556280, 3.57492036390584089042156084482, 5.12049721063222219240393062631, 6.28018464851400605090039095980, 6.60564286120665850318056002212, 8.023527167147012468724841163045, 8.729617651986676739183317895115, 9.692261486005965048110378838459, 10.43759936367809315199317138670

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