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Properties

Label 2-684-228.11-c1-0-26
Degree $2$
Conductor $684$
Sign $0.920 + 0.391i$
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  + (1.22 + 0.707i)2-s + (0.999 + 1.73i)4-s + (−2.44 − 1.41i)5-s − 4.35i·7-s + 2.82i·8-s + (−1.99 − 3.46i)10-s + 6.16·11-s + (−1.5 − 2.59i)13-s + (3.08 − 5.33i)14-s + (−2.00 + 3.46i)16-s + (2.44 + 1.41i)17-s − 4.35i·19-s − 5.65i·20-s + (7.54 + 4.35i)22-s + (1.49 + 2.59i)25-s − 4.24i·26-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)2-s + (0.499 + 0.866i)4-s + (−1.09 − 0.632i)5-s − 1.64i·7-s + 0.999i·8-s + (−0.632 − 1.09i)10-s + 1.85·11-s + (−0.416 − 0.720i)13-s + (0.823 − 1.42i)14-s + (−0.500 + 0.866i)16-s + (0.594 + 0.342i)17-s − 0.999i·19-s − 1.26i·20-s + (1.60 + 0.929i)22-s + (0.299 + 0.519i)25-s − 0.832i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.920 + 0.391i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ 0.920 + 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11631 - 0.431892i\)
\(L(\frac12)\) \(\approx\) \(2.11631 - 0.431892i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 \)
19 \( 1 + 4.35iT \)
good5 \( 1 + (2.44 + 1.41i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 4.35iT - 7T^{2} \)
11 \( 1 - 6.16T + 11T^{2} \)
13 \( 1 + (1.5 + 2.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.44 - 1.41i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.44 + 1.41i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.35iT - 31T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 + (-4.89 - 2.82i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.77 - 2.17i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.08 + 5.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.79 - 5.65i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.08 - 5.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.5 - 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.3 - 6.53i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.08 - 5.33i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.77 + 2.17i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.3T + 83T^{2} \)
89 \( 1 + (-6.12 + 3.53i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6 + 10.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.68684561886435286936540927175, −9.466147373742153631520664629066, −8.359901645672900233531518283680, −7.58149548260702524832239774479, −7.01528486221301813378371314791, −5.99037505756802490874879690433, −4.40658810429554634463753674591, −4.28881596589969914271746196725, −3.24189021991303992514274176653, −0.964305107372604920570031814138, 1.74170012854921414916247598393, 3.06099149787832825655764694186, 3.84672710122875734917660340371, 4.90078555550564760556839148056, 6.09377481550933055165586898477, 6.69165393601165172590094909325, 7.82100519336046922686323351110, 9.075277180846021473268807812148, 9.620236578747910456188688848300, 10.95519409246204550776445961819

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