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Properties

Label 2-684-76.3-c1-0-3
Degree $2$
Conductor $684$
Sign $-0.850 - 0.526i$
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  + (−1.39 − 0.250i)2-s + (1.87 + 0.697i)4-s + (−0.683 + 3.87i)5-s + (0.181 − 0.104i)7-s + (−2.43 − 1.44i)8-s + (1.92 − 5.22i)10-s + (−0.749 − 0.432i)11-s + (−0.989 + 2.71i)13-s + (−0.279 + 0.100i)14-s + (3.02 + 2.61i)16-s + (2.09 + 1.75i)17-s + (−0.105 + 4.35i)19-s + (−3.98 + 6.78i)20-s + (0.935 + 0.790i)22-s + (2.73 − 0.482i)23-s + ⋯
L(s)  = 1  + (−0.984 − 0.177i)2-s + (0.937 + 0.348i)4-s + (−0.305 + 1.73i)5-s + (0.0686 − 0.0396i)7-s + (−0.860 − 0.509i)8-s + (0.607 − 1.65i)10-s + (−0.226 − 0.130i)11-s + (−0.274 + 0.753i)13-s + (−0.0745 + 0.0268i)14-s + (0.756 + 0.653i)16-s + (0.508 + 0.426i)17-s + (−0.0241 + 0.999i)19-s + (−0.890 + 1.51i)20-s + (0.199 + 0.168i)22-s + (0.570 − 0.100i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.850 - 0.526i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ -0.850 - 0.526i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.155201 + 0.545723i\)
\(L(\frac12)\) \(\approx\) \(0.155201 + 0.545723i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.250i)T \)
3 \( 1 \)
19 \( 1 + (0.105 - 4.35i)T \)
good5 \( 1 + (0.683 - 3.87i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (-0.181 + 0.104i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.749 + 0.432i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.989 - 2.71i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-2.09 - 1.75i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-2.73 + 0.482i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (5.73 + 6.83i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (2.99 + 5.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.151iT - 37T^{2} \)
41 \( 1 + (-1.95 - 5.36i)T + (-31.4 + 26.3i)T^{2} \)
43 \( 1 + (10.1 + 1.79i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (-2.13 - 2.54i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (9.34 - 1.64i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (-2.89 - 2.43i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.46 - 8.30i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (7.42 - 6.23i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-2.01 + 11.4i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-6.12 + 2.23i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-16.0 + 5.85i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (13.7 - 7.93i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.622 - 1.71i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (6.40 - 7.62i)T + (-16.8 - 95.5i)T^{2} \)
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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

−10.76797687128211689442473217891, −10.03075240378217066872434829068, −9.343278516782100962084173913271, −7.964636739817931593627931842878, −7.57720288285684233918837954698, −6.60226141002982237471499200631, −5.92593849617120605338215988762, −3.94185545217000665692145709874, −3.02447828013888614684717911368, −1.94304053259237963393093913321, 0.40402686847984828120530133952, 1.65993704902053255937406452110, 3.30326738779949084521786733545, 5.06102311151218021966810427641, 5.32867664104398154947452909645, 6.89214019612111088496111527387, 7.73541753271193243661831325546, 8.524131725948532836539876074290, 9.124125342823379591921754305303, 9.844719568506789610029836722256

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