Properties

Label 2-684-19.7-c1-0-5
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  + (−0.5 − 0.866i)5-s + 4·11-s + (0.5 − 0.866i)13-s + (1.5 + 2.59i)17-s + (−4 − 1.73i)19-s + (2.5 − 4.33i)23-s + (2 − 3.46i)25-s + (3.5 − 6.06i)29-s + 4·31-s + 10·37-s + (−2.5 − 4.33i)41-s + (2.5 + 4.33i)43-s + (−3.5 + 6.06i)47-s − 7·49-s + (5.5 − 9.52i)53-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + 1.20·11-s + (0.138 − 0.240i)13-s + (0.363 + 0.630i)17-s + (−0.917 − 0.397i)19-s + (0.521 − 0.902i)23-s + (0.400 − 0.692i)25-s + (0.649 − 1.12i)29-s + 0.718·31-s + 1.64·37-s + (−0.390 − 0.676i)41-s + (0.381 + 0.660i)43-s + (−0.510 + 0.884i)47-s − 49-s + (0.755 − 1.30i)53-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: $\chi_{684} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ 0.813 + 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45338 - 0.466753i\)
\(L(\frac12)\) \(\approx\) \(1.45338 - 0.466753i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.5 + 4.33i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.5 + 6.06i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.5 - 4.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.5 - 6.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.5 + 9.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.5 - 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.5 - 9.52i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.5 + 12.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.39055879973203081217386114195, −9.497351295310734943671927488449, −8.580652816347476968760742032571, −8.015615020936770040446543523647, −6.65806425345277821124763210627, −6.12993635719395679725819795471, −4.68464939759918247971089997218, −4.02060887575246387022468094790, −2.59662526913971519372896036577, −0.977606918264635694411434422851, 1.37326365594736512890603353636, 2.98585148995980340121253393424, 3.96870431994406708542770645362, 5.05780718076940847184524796367, 6.30556778419459815176130522348, 6.94265622070160731094303134763, 7.933657381882974717275263284623, 8.952138043441643345854234310768, 9.611260232005024940752460231106, 10.65176902643979957407562522284

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