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Properties

Label 2-684-12.11-c1-0-11
Degree $2$
Conductor $684$
Sign $-0.520 - 0.853i$
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  + (0.965 + 1.03i)2-s + (−0.136 + 1.99i)4-s + 0.0590i·5-s + 1.27i·7-s + (−2.19 + 1.78i)8-s + (−0.0609 + 0.0569i)10-s + 5.27·11-s − 1.40·13-s + (−1.31 + 1.22i)14-s + (−3.96 − 0.543i)16-s + 8.20i·17-s − i·19-s + (−0.117 − 0.00803i)20-s + (5.09 + 5.45i)22-s − 7.75·23-s + ⋯
L(s)  = 1  + (0.682 + 0.730i)2-s + (−0.0680 + 0.997i)4-s + 0.0263i·5-s + 0.480i·7-s + (−0.775 + 0.631i)8-s + (−0.0192 + 0.0180i)10-s + 1.59·11-s − 0.389·13-s + (−0.351 + 0.328i)14-s + (−0.990 − 0.135i)16-s + 1.98i·17-s − 0.229i·19-s + (−0.0263 − 0.00179i)20-s + (1.08 + 1.16i)22-s − 1.61·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.520 - 0.853i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ -0.520 - 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01006 + 1.79849i\)
\(L(\frac12)\) \(\approx\) \(1.01006 + 1.79849i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 1.03i)T \)
3 \( 1 \)
19 \( 1 + iT \)
good5 \( 1 - 0.0590iT - 5T^{2} \)
7 \( 1 - 1.27iT - 7T^{2} \)
11 \( 1 - 5.27T + 11T^{2} \)
13 \( 1 + 1.40T + 13T^{2} \)
17 \( 1 - 8.20iT - 17T^{2} \)
23 \( 1 + 7.75T + 23T^{2} \)
29 \( 1 - 6.17iT - 29T^{2} \)
31 \( 1 + 7.05iT - 31T^{2} \)
37 \( 1 + 5.75T + 37T^{2} \)
41 \( 1 - 7.99iT - 41T^{2} \)
43 \( 1 + 4.49iT - 43T^{2} \)
47 \( 1 - 6.31T + 47T^{2} \)
53 \( 1 + 10.5iT - 53T^{2} \)
59 \( 1 - 6.05T + 59T^{2} \)
61 \( 1 - 6.72T + 61T^{2} \)
67 \( 1 + 2.89iT - 67T^{2} \)
71 \( 1 - 8.96T + 71T^{2} \)
73 \( 1 + 3.05T + 73T^{2} \)
79 \( 1 + 2.67iT - 79T^{2} \)
83 \( 1 - 9.34T + 83T^{2} \)
89 \( 1 + 2.25iT - 89T^{2} \)
97 \( 1 + 14.2T + 97T^{2} \)
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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

−10.92911168417430037819595520844, −9.748579432958965679796349976039, −8.750733740106604508795173155638, −8.223232021772387015318974122510, −6.98920490915082724588046235380, −6.30982494698548566077027663093, −5.52012604782111774580432087033, −4.25541612652027519045753954504, −3.57832039926286955435707832506, −2.00543040567481248099413704317, 0.928918189863402331457931477849, 2.38881739796318963058428846136, 3.66003186648569952313700254040, 4.44356250620458850228758209654, 5.46634926294780152426712945556, 6.58700675927523261662913426364, 7.27390831126604579162831502612, 8.792284271913867348976035308905, 9.529748784736988700905112703381, 10.25095386258387103161860774674

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