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Properties

Label 2-684-76.67-c1-0-2
Degree $2$
Conductor $684$
Sign $-0.460 - 0.887i$
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  + (−1.33 − 0.459i)2-s + (1.57 + 1.22i)4-s + (2.14 + 0.779i)5-s + (−3.55 − 2.04i)7-s + (−1.54 − 2.36i)8-s + (−2.50 − 2.02i)10-s + (−3.61 + 2.08i)11-s + (−0.374 + 0.446i)13-s + (3.80 + 4.37i)14-s + (0.982 + 3.87i)16-s + (0.573 + 3.25i)17-s + (−0.458 + 4.33i)19-s + (2.42 + 3.85i)20-s + (5.80 − 1.13i)22-s + (0.862 + 2.37i)23-s + ⋯
L(s)  = 1  + (−0.945 − 0.324i)2-s + (0.789 + 0.614i)4-s + (0.957 + 0.348i)5-s + (−1.34 − 0.774i)7-s + (−0.547 − 0.837i)8-s + (−0.792 − 0.640i)10-s + (−1.09 + 0.630i)11-s + (−0.103 + 0.123i)13-s + (1.01 + 1.16i)14-s + (0.245 + 0.969i)16-s + (0.138 + 0.788i)17-s + (−0.105 + 0.994i)19-s + (0.541 + 0.862i)20-s + (1.23 − 0.241i)22-s + (0.179 + 0.494i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.460 - 0.887i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (523, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ -0.460 - 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.217856 + 0.358323i\)
\(L(\frac12)\) \(\approx\) \(0.217856 + 0.358323i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 + 0.459i)T \)
3 \( 1 \)
19 \( 1 + (0.458 - 4.33i)T \)
good5 \( 1 + (-2.14 - 0.779i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (3.55 + 2.04i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.61 - 2.08i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.374 - 0.446i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.573 - 3.25i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-0.862 - 2.37i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (8.34 + 1.47i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (0.386 - 0.670i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.23iT - 37T^{2} \)
41 \( 1 + (-4.51 - 5.37i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (1.55 - 4.27i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (4.84 + 0.854i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-0.232 - 0.639i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (-0.368 - 2.08i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (2.91 - 1.05i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-2.44 + 13.8i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-12.4 - 4.51i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (8.59 - 7.20i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (7.91 - 6.63i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (1.29 + 0.747i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (9.52 - 11.3i)T + (-15.4 - 87.6i)T^{2} \)
97 \( 1 + (-10.1 + 1.78i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.42416518565966859101400997466, −9.822836886221967285655488497734, −9.545772143804947233665253307298, −8.132509433932349553509732718029, −7.35512521535977535443916473297, −6.48212288475332197348521361685, −5.72382642059063538734845320515, −3.91103347697617397771308540982, −2.88319784369919886184005556204, −1.73568225554566499438996567216, 0.27462775792372801053390827594, 2.22971136159202489411721441634, 3.04653180563394141472932056150, 5.28749423179571454724213289273, 5.73963561279344953152159235817, 6.67241420364978986217374355356, 7.60025570774952148157231928625, 8.832102336730941663921050219599, 9.239816400534490106624847725130, 9.947378239164452297191600006312

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