Properties

Label 2-684-3.2-c2-0-6
Degree $2$
Conductor $684$
Sign $0.577 + 0.816i$
Analytic cond. $18.6376$
Root an. cond. $4.31713$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.16i·5-s − 9.11·7-s + 1.10i·11-s + 2.28·13-s − 19.2i·17-s + 4.35·19-s − 1.28i·23-s + 20.3·25-s − 37.0i·29-s + 31.3·31-s − 19.6i·35-s + 34.9·37-s − 43.5i·41-s + 0.553·43-s + 9.35i·47-s + ⋯
L(s)  = 1  + 0.432i·5-s − 1.30·7-s + 0.100i·11-s + 0.175·13-s − 1.13i·17-s + 0.229·19-s − 0.0559i·23-s + 0.813·25-s − 1.27i·29-s + 1.01·31-s − 0.562i·35-s + 0.944·37-s − 1.06i·41-s + 0.0128·43-s + 0.198i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ 0.577 + 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.277372050\)
\(L(\frac12)\) \(\approx\) \(1.277372050\)
\(L(2)\) 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 \( 1 - 4.35T \)
good5 \( 1 - 2.16iT - 25T^{2} \)
7 \( 1 + 9.11T + 49T^{2} \)
11 \( 1 - 1.10iT - 121T^{2} \)
13 \( 1 - 2.28T + 169T^{2} \)
17 \( 1 + 19.2iT - 289T^{2} \)
23 \( 1 + 1.28iT - 529T^{2} \)
29 \( 1 + 37.0iT - 841T^{2} \)
31 \( 1 - 31.3T + 961T^{2} \)
37 \( 1 - 34.9T + 1.36e3T^{2} \)
41 \( 1 + 43.5iT - 1.68e3T^{2} \)
43 \( 1 - 0.553T + 1.84e3T^{2} \)
47 \( 1 - 9.35iT - 2.20e3T^{2} \)
53 \( 1 + 75.0iT - 2.80e3T^{2} \)
59 \( 1 - 38.1iT - 3.48e3T^{2} \)
61 \( 1 + 62.4T + 3.72e3T^{2} \)
67 \( 1 - 85.6T + 4.48e3T^{2} \)
71 \( 1 - 15.9iT - 5.04e3T^{2} \)
73 \( 1 - 67.1T + 5.32e3T^{2} \)
79 \( 1 + 58.5T + 6.24e3T^{2} \)
83 \( 1 + 115. iT - 6.88e3T^{2} \)
89 \( 1 - 6.59iT - 7.92e3T^{2} \)
97 \( 1 + 14.0T + 9.40e3T^{2} \)
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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

−9.939528242158654401550755813256, −9.542399759547575017612591018594, −8.470445350785701232388700876610, −7.33392195854315690240125475088, −6.63701356538280576136145123204, −5.81484536642604363890690704010, −4.55592608107898268242146099344, −3.34655469943449665573292401591, −2.52723009891828486767925363507, −0.53613223488668531145845719864, 1.11665968961109240194198091715, 2.81376600500207418615330310219, 3.77414241343977802236440826356, 4.93474010274908544027794017507, 6.09999211673983379638782437901, 6.68292175542382966478021487273, 7.88316039554588672969935031418, 8.779262405612190657597714035295, 9.527697320997277551150712169672, 10.33118556841104867834410896235

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