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Properties

Label 2-684-684.391-c0-0-0
Degree $2$
Conductor $684$
Sign $-0.790 - 0.612i$
Analytic cond. $0.341360$
Root an. cond. $0.584260$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + i·3-s + 4-s + (0.5 + 0.866i)5-s − i·6-s + (−0.866 + 0.5i)7-s − 8-s − 9-s + (−0.5 − 0.866i)10-s + (−0.866 + 0.5i)11-s + i·12-s + (0.866 − 0.5i)14-s + (−0.866 + 0.5i)15-s + 16-s + (−0.5 + 0.866i)17-s + 18-s + ⋯
L(s)  = 1  − 2-s + i·3-s + 4-s + (0.5 + 0.866i)5-s − i·6-s + (−0.866 + 0.5i)7-s − 8-s − 9-s + (−0.5 − 0.866i)10-s + (−0.866 + 0.5i)11-s + i·12-s + (0.866 − 0.5i)14-s + (−0.866 + 0.5i)15-s + 16-s + (−0.5 + 0.866i)17-s + 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.790 - 0.612i$
Analytic conductor: \(0.341360\)
Root analytic conductor: \(0.584260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :0),\ -0.790 - 0.612i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4935645889\)
\(L(\frac12)\) \(\approx\) \(0.4935645889\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - iT \)
19 \( 1 + iT \)
good5 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 - 2iT - T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−10.68717330499323686551662273747, −10.08143246556838853664608620271, −9.526046609247441017411484279373, −8.693697400824422621501517730094, −7.72644696688611407459642975456, −6.43553914799520592070323145407, −6.07730700663888728028844693879, −4.64304014968089064762369416177, −2.99621040443617869501431362963, −2.52170276451236376713163624792, 0.70939429206696230069454121329, 2.13082038791827503039043514562, 3.28442315299196030428963039769, 5.27763592596397861649505749196, 6.12134011226826594357168109103, 6.99620138376644205952342791128, 7.79070501573433649415662561551, 8.663439572792869600934722800138, 9.306122018403627528657083133621, 10.27752631201605657421330520872

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