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Properties

Label 2-684-19.11-c1-0-1
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  − 7-s + (3.5 + 6.06i)13-s + (4 − 1.73i)19-s + (2.5 + 4.33i)25-s + 11·31-s − 37-s + (−2.5 + 4.33i)43-s − 6·49-s + (6.5 + 11.2i)61-s + (−2.5 − 4.33i)67-s + (3.5 − 6.06i)73-s + (6.5 − 11.2i)79-s + (−3.5 − 6.06i)91-s + (−7 + 12.1i)97-s − 13·103-s + ⋯
L(s)  = 1  − 0.377·7-s + (0.970 + 1.68i)13-s + (0.917 − 0.397i)19-s + (0.5 + 0.866i)25-s + 1.97·31-s − 0.164·37-s + (−0.381 + 0.660i)43-s − 0.857·49-s + (0.832 + 1.44i)61-s + (−0.305 − 0.529i)67-s + (0.409 − 0.709i)73-s + (0.731 − 1.26i)79-s + (−0.366 − 0.635i)91-s + (−0.710 + 1.23i)97-s − 1.28·103-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

\(L(1)\) \(\approx\) \(1.41554 + 0.454602i\)
\(L(\frac12)\) \(\approx\) \(1.41554 + 0.454602i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-3.5 - 6.06i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 11T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7 - 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.58879529167237870064069907125, −9.548448162232928792631016333427, −9.002651724365164864793080192360, −8.006174992979102293291042449070, −6.86417724246588241096283935443, −6.31109773730481798330143354615, −5.05159542177268024693787578947, −4.03767499036298439738601663654, −2.92436734698993218876634694339, −1.38182999948179400056795663445, 0.931201664825024208370380336841, 2.78396680113423007614359943982, 3.66299249576563919500134900564, 5.01504930672454926815802982300, 5.92610215535546317499030056157, 6.77188411045087413249770336375, 8.009888937858998171785173582196, 8.451295298584189737520893350974, 9.738899619383556626914400116752, 10.29493081965397001916050512431

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