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Properties

Label 2-684-19.15-c0-0-0
Degree $2$
Conductor $684$
Sign $0.992 + 0.120i$
Analytic cond. $0.341360$
Root an. cond. $0.584260$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 − 0.300i)7-s + (1.26 + 0.223i)13-s + (0.5 − 0.866i)19-s + (−0.173 + 0.984i)25-s + (−0.592 − 0.342i)31-s − 0.684i·37-s + (−1.43 + 1.20i)43-s + (0.439 + 0.761i)49-s + (−1.17 − 0.984i)61-s + (−0.673 − 1.85i)67-s + (0.266 + 1.50i)73-s + (−1.93 + 0.342i)79-s + (0.286 − 0.342i)91-s + (−0.592 + 1.62i)97-s + (−1.11 + 0.642i)103-s + ⋯
L(s)  = 1  + (0.173 − 0.300i)7-s + (1.26 + 0.223i)13-s + (0.5 − 0.866i)19-s + (−0.173 + 0.984i)25-s + (−0.592 − 0.342i)31-s − 0.684i·37-s + (−1.43 + 1.20i)43-s + (0.439 + 0.761i)49-s + (−1.17 − 0.984i)61-s + (−0.673 − 1.85i)67-s + (0.266 + 1.50i)73-s + (−1.93 + 0.342i)79-s + (0.286 − 0.342i)91-s + (−0.592 + 1.62i)97-s + (−1.11 + 0.642i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.992 + 0.120i$
Analytic conductor: \(0.341360\)
Root analytic conductor: \(0.584260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :0),\ 0.992 + 0.120i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9934736495\)
\(L(\frac12)\) \(\approx\) \(0.9934736495\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (0.173 - 0.984i)T^{2} \)
7 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.26 - 0.223i)T + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.766 + 0.642i)T^{2} \)
23 \( 1 + (0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (0.592 + 0.342i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + 0.684iT - T^{2} \)
41 \( 1 + (0.939 - 0.342i)T^{2} \)
43 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (0.766 - 0.642i)T^{2} \)
53 \( 1 + (-0.173 - 0.984i)T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2} \)
67 \( 1 + (0.673 + 1.85i)T + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
79 \( 1 + (1.93 - 0.342i)T + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.939 + 0.342i)T^{2} \)
97 \( 1 + (0.592 - 1.62i)T + (-0.766 - 0.642i)T^{2} \)
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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

−10.99292849380964429654084901097, −9.704030326828789823234205827512, −9.020117991550762485058564663836, −8.067705491979884392212325146904, −7.18468926952169906891068501084, −6.24709248013325533580793953248, −5.24878685478937538249821368770, −4.13736558675381106353360772261, −3.11232544877485692927436509502, −1.49219031819791772022830470178, 1.60769552276505894918811906902, 3.12840136531938841681373630438, 4.15150028260180327291670876234, 5.41229490700873079074749976410, 6.15445842382711140112471029810, 7.21501617772971098462621695038, 8.295767936786091469123624481403, 8.789692427325710914689404402639, 9.987108360232228153659880026543, 10.60826526278565807621625997636

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