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Properties

Label 2-684-171.106-c1-0-9
Degree $2$
Conductor $684$
Sign $0.0408 + 0.999i$
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  + (−1.04 + 1.38i)3-s − 2.40·5-s + (−1.53 + 2.65i)7-s + (−0.810 − 2.88i)9-s + (−2.96 + 5.14i)11-s + (2.81 − 4.88i)13-s + (2.51 − 3.32i)15-s + (1.15 − 2.00i)17-s + (1.16 − 4.19i)19-s + (−2.05 − 4.89i)21-s + (−0.654 + 1.13i)23-s + 0.787·25-s + (4.83 + 1.90i)27-s + 6.77·29-s + (−4.86 − 8.42i)31-s + ⋯
L(s)  = 1  + (−0.604 + 0.796i)3-s − 1.07·5-s + (−0.578 + 1.00i)7-s + (−0.270 − 0.962i)9-s + (−0.894 + 1.55i)11-s + (0.782 − 1.35i)13-s + (0.649 − 0.857i)15-s + (0.280 − 0.485i)17-s + (0.267 − 0.963i)19-s + (−0.449 − 1.06i)21-s + (−0.136 + 0.236i)23-s + 0.157·25-s + (0.930 + 0.366i)27-s + 1.25·29-s + (−0.873 − 1.51i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0408 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0408 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.0408 + 0.999i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ 0.0408 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.172339 - 0.165436i\)
\(L(\frac12)\) \(\approx\) \(0.172339 - 0.165436i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.04 - 1.38i)T \)
19 \( 1 + (-1.16 + 4.19i)T \)
good5 \( 1 + 2.40T + 5T^{2} \)
7 \( 1 + (1.53 - 2.65i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.96 - 5.14i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.81 + 4.88i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.15 + 2.00i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.654 - 1.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.77T + 29T^{2} \)
31 \( 1 + (4.86 + 8.42i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.91T + 37T^{2} \)
41 \( 1 + 3.91T + 41T^{2} \)
43 \( 1 + (0.0602 + 0.104i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 + (4.85 + 8.41i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.36T + 59T^{2} \)
61 \( 1 - 9.94T + 61T^{2} \)
67 \( 1 + (-5.03 + 8.72i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.00 - 1.73i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.09 - 7.10i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.70 + 9.88i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.77 - 4.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.48 + 7.76i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.43 + 5.94i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.14626090537871023743015127011, −9.667694441241046823229402917963, −8.560322336303508901088171856978, −7.71492172743610996736249665567, −6.67563433049463548615798224008, −5.50216074994540392922269653028, −4.89211453342603558962411754844, −3.70318661592580024131849984954, −2.73275068356261620783521161849, −0.15036194332776219697060408211, 1.26216060605804121714939882419, 3.24076092912795527119664760624, 4.06007225616521542944856898926, 5.41459274234205605577551542564, 6.44532378597224373293675327304, 7.07120807693670211844046121351, 8.135555735470160378796431540586, 8.511593728947381032890867110032, 10.17401362077193193015993888724, 10.84979824646347076765435118927

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