Properties

Label 2-684-4.3-c2-0-5
Degree $2$
Conductor $684$
Sign $-0.965 - 0.258i$
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  + (−1.58 + 1.21i)2-s + (1.03 − 3.86i)4-s + 1.46·5-s − 6.38i·7-s + (3.06 + 7.39i)8-s + (−2.32 + 1.78i)10-s − 9.33i·11-s − 22.7·13-s + (7.77 + 10.1i)14-s + (−13.8 − 8.00i)16-s − 5.17·17-s − 4.35i·19-s + (1.52 − 5.67i)20-s + (11.3 + 14.8i)22-s + 37.0i·23-s + ⋯
L(s)  = 1  + (−0.793 + 0.608i)2-s + (0.258 − 0.965i)4-s + 0.293·5-s − 0.912i·7-s + (0.382 + 0.923i)8-s + (−0.232 + 0.178i)10-s − 0.849i·11-s − 1.74·13-s + (0.555 + 0.724i)14-s + (−0.866 − 0.500i)16-s − 0.304·17-s − 0.229i·19-s + (0.0760 − 0.283i)20-s + (0.516 + 0.673i)22-s + 1.61i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.965 - 0.258i$
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: \((2,\ 684,\ (\ :1),\ -0.965 - 0.258i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1948192171\)
\(L(\frac12)\) \(\approx\) \(0.1948192171\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.58 - 1.21i)T \)
3 \( 1 \)
19 \( 1 + 4.35iT \)
good5 \( 1 - 1.46T + 25T^{2} \)
7 \( 1 + 6.38iT - 49T^{2} \)
11 \( 1 + 9.33iT - 121T^{2} \)
13 \( 1 + 22.7T + 169T^{2} \)
17 \( 1 + 5.17T + 289T^{2} \)
23 \( 1 - 37.0iT - 529T^{2} \)
29 \( 1 - 23.0T + 841T^{2} \)
31 \( 1 - 19.9iT - 961T^{2} \)
37 \( 1 - 24.3T + 1.36e3T^{2} \)
41 \( 1 + 57.3T + 1.68e3T^{2} \)
43 \( 1 - 78.8iT - 1.84e3T^{2} \)
47 \( 1 - 50.2iT - 2.20e3T^{2} \)
53 \( 1 - 82.4T + 2.80e3T^{2} \)
59 \( 1 - 86.4iT - 3.48e3T^{2} \)
61 \( 1 + 114.T + 3.72e3T^{2} \)
67 \( 1 + 64.7iT - 4.48e3T^{2} \)
71 \( 1 + 25.1iT - 5.04e3T^{2} \)
73 \( 1 + 58.1T + 5.32e3T^{2} \)
79 \( 1 + 47.6iT - 6.24e3T^{2} \)
83 \( 1 - 51.5iT - 6.88e3T^{2} \)
89 \( 1 - 4.37T + 7.92e3T^{2} \)
97 \( 1 - 16.1T + 9.40e3T^{2} \)
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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

−10.38503515049668232288865913465, −9.767666165084071605680970032233, −9.034898709445464702902429997723, −7.86424217601747321547616429909, −7.36472231856012591422029926489, −6.41364314422782494666483666300, −5.43609125643502683156416688802, −4.46414318101830850451733247724, −2.83538442516177824967623264574, −1.34732316129551013934105171865, 0.089552896539637301321629034116, 2.08877072315929092942131885891, 2.56947873375738428582583866479, 4.19120761523446376612240633485, 5.22290035928776300505558990979, 6.57651631810090441070895219602, 7.38140852177736904167374749437, 8.367230150471796474240964555476, 9.132215000171452575026500377223, 9.998743263294264973739514698685

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