Properties

Label 2-684-4.3-c2-0-83
Degree $2$
Conductor $684$
Sign $-0.999 + 0.0355i$
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  + (1.38 − 1.43i)2-s + (−0.142 − 3.99i)4-s − 1.77·5-s − 3.79i·7-s + (−5.94 − 5.34i)8-s + (−2.46 + 2.55i)10-s − 9.58i·11-s + 22.4·13-s + (−5.45 − 5.26i)14-s + (−15.9 + 1.13i)16-s − 11.9·17-s + 4.35i·19-s + (0.252 + 7.09i)20-s + (−13.7 − 13.3i)22-s − 30.8i·23-s + ⋯
L(s)  = 1  + (0.694 − 0.719i)2-s + (−0.0355 − 0.999i)4-s − 0.355·5-s − 0.541i·7-s + (−0.743 − 0.668i)8-s + (−0.246 + 0.255i)10-s − 0.871i·11-s + 1.72·13-s + (−0.389 − 0.376i)14-s + (−0.997 + 0.0709i)16-s − 0.705·17-s + 0.229i·19-s + (0.0126 + 0.354i)20-s + (−0.626 − 0.604i)22-s − 1.34i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0355i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.999 + 0.0355i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ -0.999 + 0.0355i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.794046113\)
\(L(\frac12)\) \(\approx\) \(1.794046113\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 1.43i)T \)
3 \( 1 \)
19 \( 1 - 4.35iT \)
good5 \( 1 + 1.77T + 25T^{2} \)
7 \( 1 + 3.79iT - 49T^{2} \)
11 \( 1 + 9.58iT - 121T^{2} \)
13 \( 1 - 22.4T + 169T^{2} \)
17 \( 1 + 11.9T + 289T^{2} \)
23 \( 1 + 30.8iT - 529T^{2} \)
29 \( 1 + 36.2T + 841T^{2} \)
31 \( 1 - 40.0iT - 961T^{2} \)
37 \( 1 + 58.0T + 1.36e3T^{2} \)
41 \( 1 + 14.5T + 1.68e3T^{2} \)
43 \( 1 + 34.2iT - 1.84e3T^{2} \)
47 \( 1 + 41.1iT - 2.20e3T^{2} \)
53 \( 1 + 22.2T + 2.80e3T^{2} \)
59 \( 1 + 62.8iT - 3.48e3T^{2} \)
61 \( 1 - 56.7T + 3.72e3T^{2} \)
67 \( 1 + 79.2iT - 4.48e3T^{2} \)
71 \( 1 - 127. iT - 5.04e3T^{2} \)
73 \( 1 - 107.T + 5.32e3T^{2} \)
79 \( 1 + 99.0iT - 6.24e3T^{2} \)
83 \( 1 + 18.4iT - 6.88e3T^{2} \)
89 \( 1 - 50.7T + 7.92e3T^{2} \)
97 \( 1 + 74.6T + 9.40e3T^{2} \)
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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

−10.27136166144445740618460951609, −8.935807465846547846735922214710, −8.388682374956643833036026105229, −6.92150476944376581498242426098, −6.15239325754943551458750085876, −5.16276958564050107480490868743, −3.88607166741145991339215485044, −3.47649279334430270434337584341, −1.85785904634471732407524881451, −0.49542758251978325368722225432, 1.99703313677000658388040269249, 3.49584327508304106662875895971, 4.21752040308982983958103019317, 5.43126535612959139520793491291, 6.13766761526887142774261130689, 7.15007780501681619471144446084, 7.940575383715870018657075995905, 8.819345592466460585180445039536, 9.565618379932248952205009919860, 11.07797322825168056924149946499

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