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Properties

Label 2-684-684.535-c1-0-82
Degree $2$
Conductor $684$
Sign $-0.605 + 0.796i$
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  + (−0.535 − 1.30i)2-s + (1.73 + 0.0149i)3-s + (−1.42 + 1.40i)4-s + (−0.784 − 0.658i)5-s + (−0.907 − 2.27i)6-s + (−0.920 − 0.531i)7-s + (2.59 + 1.11i)8-s + (2.99 + 0.0518i)9-s + (−0.441 + 1.37i)10-s − 1.95i·11-s + (−2.49 + 2.40i)12-s + (−3.18 − 3.79i)13-s + (−0.203 + 1.48i)14-s + (−1.34 − 1.15i)15-s + (0.0725 − 3.99i)16-s + (2.18 + 1.83i)17-s + ⋯
L(s)  = 1  + (−0.378 − 0.925i)2-s + (0.999 + 0.00864i)3-s + (−0.713 + 0.700i)4-s + (−0.350 − 0.294i)5-s + (−0.370 − 0.928i)6-s + (−0.348 − 0.200i)7-s + (0.918 + 0.395i)8-s + (0.999 + 0.0172i)9-s + (−0.139 + 0.436i)10-s − 0.588i·11-s + (−0.719 + 0.694i)12-s + (−0.882 − 1.05i)13-s + (−0.0542 + 0.398i)14-s + (−0.348 − 0.297i)15-s + (0.0181 − 0.999i)16-s + (0.530 + 0.445i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.605 + 0.796i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ -0.605 + 0.796i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.600895 - 1.21179i\)
\(L(\frac12)\) \(\approx\) \(0.600895 - 1.21179i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.535 + 1.30i)T \)
3 \( 1 + (-1.73 - 0.0149i)T \)
19 \( 1 + (-2.84 + 3.30i)T \)
good5 \( 1 + (0.784 + 0.658i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (0.920 + 0.531i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 1.95iT - 11T^{2} \)
13 \( 1 + (3.18 + 3.79i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (-2.18 - 1.83i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-0.416 + 1.14i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.544 + 1.49i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + 3.11T + 31T^{2} \)
37 \( 1 + 8.94iT - 37T^{2} \)
41 \( 1 + (1.61 + 0.284i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-2.23 - 6.14i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.96 - 5.40i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (-1.47 + 0.259i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (-9.74 + 3.54i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-5.18 + 4.34i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-1.37 - 7.80i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (1.76 - 9.99i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (2.87 - 1.04i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-0.624 - 0.524i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (2.89 + 1.67i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.89 - 7.94i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (8.30 + 1.46i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.02338386830774807845755099043, −9.463381074652961425330477608477, −8.471788973860394288794753331147, −7.926845754785439173235766574047, −7.08234813273510911696256422021, −5.34889631918509200883743454160, −4.19442341349896021161382180629, −3.29608953240767264114756933190, −2.44122728277832668885923017701, −0.75791940991448097699205700224, 1.74161928456859397291277872541, 3.28560212632131903618218331610, 4.38088928409615811545519789781, 5.41341896489432222310333917258, 6.84972696907657476856091534260, 7.26400833760415956594320467092, 8.062743042703325864813727133688, 9.077586424893242679628604656045, 9.663750655228679813293584717696, 10.25195544460606180586716893853

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