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Properties

Label 2-684-19.11-c1-0-8
Degree $2$
Conductor $684$
Sign $-0.813 + 0.582i$
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  + (2 − 3.46i)5-s − 3·7-s − 4·11-s + (−2.5 − 4.33i)13-s + (−4 + 1.73i)19-s + (2 + 3.46i)23-s + (−5.49 − 9.52i)25-s + (4 + 6.92i)29-s + 31-s + (−6 + 10.3i)35-s − 5·37-s + (4 − 6.92i)41-s + (2.5 − 4.33i)43-s + (−4 − 6.92i)47-s + 2·49-s + ⋯
L(s)  = 1  + (0.894 − 1.54i)5-s − 1.13·7-s − 1.20·11-s + (−0.693 − 1.20i)13-s + (−0.917 + 0.397i)19-s + (0.417 + 0.722i)23-s + (−1.09 − 1.90i)25-s + (0.742 + 1.28i)29-s + 0.179·31-s + (−1.01 + 1.75i)35-s − 0.821·37-s + (0.624 − 1.08i)41-s + (0.381 − 0.660i)43-s + (−0.583 − 1.01i)47-s + 0.285·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.813 + 0.582i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ -0.813 + 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.277166 - 0.863043i\)
\(L(\frac12)\) \(\approx\) \(0.277166 - 0.863043i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 + (4 - 1.73i)T \)
good5 \( 1 + (-2 + 3.46i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 3T + 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4 - 6.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + 5T + 37T^{2} \)
41 \( 1 + (-4 + 6.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.5 + 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2 - 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-8 + 13.8i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.5 + 12.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.5 + 6.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (6 + 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.14223655608284798734247874257, −9.276500448455116606976406422020, −8.562220068111626520122914371122, −7.64666404494160553927137108341, −6.39737234569397275903182508424, −5.38482928820182145468782345512, −4.99254866738420551500185459252, −3.39349154255150084750148568179, −2.14170392074495020242443445664, −0.43722654591832195116399434119, 2.43587267079603759272008604261, 2.79766505733773326127362858108, 4.30731262189271893355232930817, 5.67612067346427488099259265577, 6.67411647736059904012725342306, 6.85562235046973912094083000301, 8.172458416563539386041483765826, 9.518960336776145699065924763254, 9.918592709230976737513832177353, 10.67910318619161179217365685496

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