Properties

Label 2-684-228.227-c2-0-32
Degree $2$
Conductor $684$
Sign $0.988 + 0.153i$
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  + (−1.88 − 0.663i)2-s + (3.12 + 2.50i)4-s − 0.647i·5-s − 4.63i·7-s + (−4.22 − 6.79i)8-s + (−0.429 + 1.22i)10-s + 14.8·11-s + 22.0i·13-s + (−3.07 + 8.73i)14-s + (3.46 + 15.6i)16-s − 18.9i·17-s + (−1.76 + 18.9i)19-s + (1.61 − 2.01i)20-s + (−28.0 − 9.87i)22-s − 13.7·23-s + ⋯
L(s)  = 1  + (−0.943 − 0.331i)2-s + (0.780 + 0.625i)4-s − 0.129i·5-s − 0.661i·7-s + (−0.528 − 0.849i)8-s + (−0.0429 + 0.122i)10-s + 1.35·11-s + 1.69i·13-s + (−0.219 + 0.624i)14-s + (0.216 + 0.976i)16-s − 1.11i·17-s + (−0.0928 + 0.995i)19-s + (0.0809 − 0.100i)20-s + (−1.27 − 0.448i)22-s − 0.596·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.153i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.988 + 0.153i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ 0.988 + 0.153i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.244963653\)
\(L(\frac12)\) \(\approx\) \(1.244963653\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.88 + 0.663i)T \)
3 \( 1 \)
19 \( 1 + (1.76 - 18.9i)T \)
good5 \( 1 + 0.647iT - 25T^{2} \)
7 \( 1 + 4.63iT - 49T^{2} \)
11 \( 1 - 14.8T + 121T^{2} \)
13 \( 1 - 22.0iT - 169T^{2} \)
17 \( 1 + 18.9iT - 289T^{2} \)
23 \( 1 + 13.7T + 529T^{2} \)
29 \( 1 + 21.8T + 841T^{2} \)
31 \( 1 - 19.3T + 961T^{2} \)
37 \( 1 + 31.3iT - 1.36e3T^{2} \)
41 \( 1 + 38.1T + 1.68e3T^{2} \)
43 \( 1 - 32.1iT - 1.84e3T^{2} \)
47 \( 1 - 75.4T + 2.20e3T^{2} \)
53 \( 1 - 60.7T + 2.80e3T^{2} \)
59 \( 1 - 113. iT - 3.48e3T^{2} \)
61 \( 1 + 106.T + 3.72e3T^{2} \)
67 \( 1 - 25.0T + 4.48e3T^{2} \)
71 \( 1 + 67.2iT - 5.04e3T^{2} \)
73 \( 1 - 99.5T + 5.32e3T^{2} \)
79 \( 1 - 84.4T + 6.24e3T^{2} \)
83 \( 1 - 92.6T + 6.88e3T^{2} \)
89 \( 1 - 120.T + 7.92e3T^{2} \)
97 \( 1 + 140. iT - 9.40e3T^{2} \)
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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

−10.17577177711488317908193776990, −9.206660537802129134574783076267, −8.933635810480170967928515563833, −7.60175773456487714185005145889, −6.94631726255135639866115632944, −6.15583328690335447417397042549, −4.40238579223085912357726917815, −3.63669743207148988232709188488, −2.07020057367331996672854900348, −0.968510844007933616060092426975, 0.828689141176554693243473615090, 2.24320939108738237176086874025, 3.49512201162596994544571965116, 5.13355840673845940171682200222, 6.06615086670400889157844459164, 6.76796803389988900900548753679, 7.86810364306353246240270368266, 8.632051389867572942461114352231, 9.256247321821768342567543614498, 10.28958666958373713527344884989

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