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Properties

Label 2-684-171.7-c1-0-16
Degree $2$
Conductor $684$
Sign $0.887 + 0.460i$
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  + (1.23 + 1.21i)3-s + (2.12 − 3.67i)5-s + (1.38 − 2.39i)7-s + (0.0360 + 2.99i)9-s + (−1.31 + 2.26i)11-s − 0.0946·13-s + (7.09 − 1.94i)15-s + (−0.170 − 0.295i)17-s + (−1.77 − 3.97i)19-s + (4.61 − 1.26i)21-s + 4.62·23-s + (−6.51 − 11.2i)25-s + (−3.60 + 3.73i)27-s + (0.706 + 1.22i)29-s + (1.43 + 2.48i)31-s + ⋯
L(s)  = 1  + (0.711 + 0.702i)3-s + (0.949 − 1.64i)5-s + (0.522 − 0.904i)7-s + (0.0120 + 0.999i)9-s + (−0.395 + 0.684i)11-s − 0.0262·13-s + (1.83 − 0.502i)15-s + (−0.0413 − 0.0715i)17-s + (−0.408 − 0.912i)19-s + (1.00 − 0.276i)21-s + 0.963·23-s + (−1.30 − 2.25i)25-s + (−0.694 + 0.719i)27-s + (0.131 + 0.227i)29-s + (0.257 + 0.446i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.887 + 0.460i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ 0.887 + 0.460i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.20019 - 0.536865i\)
\(L(\frac12)\) \(\approx\) \(2.20019 - 0.536865i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 - 1.21i)T \)
19 \( 1 + (1.77 + 3.97i)T \)
good5 \( 1 + (-2.12 + 3.67i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.38 + 2.39i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.31 - 2.26i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.0946T + 13T^{2} \)
17 \( 1 + (0.170 + 0.295i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 - 4.62T + 23T^{2} \)
29 \( 1 + (-0.706 - 1.22i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.43 - 2.48i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 10.6T + 37T^{2} \)
41 \( 1 + (-1.21 + 2.10i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 12.5T + 43T^{2} \)
47 \( 1 + (-3.42 - 5.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.21 - 3.83i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.77 - 11.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.658 + 1.14i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 - 7.22T + 67T^{2} \)
71 \( 1 + (-5.00 - 8.67i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.76 + 8.24i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 3.53T + 79T^{2} \)
83 \( 1 + (1.93 - 3.35i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.91 - 10.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.430T + 97T^{2} \)
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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

−10.35164761089646785721156455173, −9.302048680747654237510244464632, −8.968502905001915389421720118088, −8.032860135590310545191301847535, −7.09359151136926624274195915735, −5.50596219319524333426845197137, −4.76463336532856758691372249533, −4.23336719932705701038536569466, −2.51534515430768864104264198256, −1.27226117347128732994584588463, 1.88819157180054359644833073166, 2.64139568848388728836208206072, 3.52509941636821749643471538615, 5.48575187158122205426345982473, 6.19069173796888424393635136454, 6.98759318890235949615112636826, 7.916379252272217415917307662626, 8.768474653498457231152290650075, 9.643759616142362749571224689850, 10.57867826427003021579266021326

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