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Properties

Label 2-684-4.3-c2-0-52
Degree $2$
Conductor $684$
Sign $0.529 - 0.848i$
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  + (1.92 + 0.551i)2-s + (3.39 + 2.11i)4-s + 2.32·5-s + 8.62i·7-s + (5.35 + 5.94i)8-s + (4.47 + 1.28i)10-s − 19.2i·11-s + 13.8·13-s + (−4.75 + 16.5i)14-s + (7.01 + 14.3i)16-s + 12.5·17-s + 4.35i·19-s + (7.89 + 4.93i)20-s + (10.6 − 37.0i)22-s + 37.4i·23-s + ⋯
L(s)  = 1  + (0.961 + 0.275i)2-s + (0.848 + 0.529i)4-s + 0.465·5-s + 1.23i·7-s + (0.669 + 0.743i)8-s + (0.447 + 0.128i)10-s − 1.75i·11-s + 1.06·13-s + (−0.339 + 1.18i)14-s + (0.438 + 0.898i)16-s + 0.736·17-s + 0.229i·19-s + (0.394 + 0.246i)20-s + (0.482 − 1.68i)22-s + 1.63i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.529 - 0.848i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.529 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.529 - 0.848i$
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: \((2,\ 684,\ (\ :1),\ 0.529 - 0.848i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.921807325\)
\(L(\frac12)\) \(\approx\) \(3.921807325\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.92 - 0.551i)T \)
3 \( 1 \)
19 \( 1 - 4.35iT \)
good5 \( 1 - 2.32T + 25T^{2} \)
7 \( 1 - 8.62iT - 49T^{2} \)
11 \( 1 + 19.2iT - 121T^{2} \)
13 \( 1 - 13.8T + 169T^{2} \)
17 \( 1 - 12.5T + 289T^{2} \)
23 \( 1 - 37.4iT - 529T^{2} \)
29 \( 1 - 6.36T + 841T^{2} \)
31 \( 1 + 5.44iT - 961T^{2} \)
37 \( 1 - 20.9T + 1.36e3T^{2} \)
41 \( 1 + 72.8T + 1.68e3T^{2} \)
43 \( 1 - 10.1iT - 1.84e3T^{2} \)
47 \( 1 - 32.4iT - 2.20e3T^{2} \)
53 \( 1 - 42.9T + 2.80e3T^{2} \)
59 \( 1 + 38.9iT - 3.48e3T^{2} \)
61 \( 1 - 25.5T + 3.72e3T^{2} \)
67 \( 1 + 65.3iT - 4.48e3T^{2} \)
71 \( 1 - 18.7iT - 5.04e3T^{2} \)
73 \( 1 - 72.8T + 5.32e3T^{2} \)
79 \( 1 + 139. iT - 6.24e3T^{2} \)
83 \( 1 + 94.7iT - 6.88e3T^{2} \)
89 \( 1 + 33.3T + 7.92e3T^{2} \)
97 \( 1 + 150.T + 9.40e3T^{2} \)
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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

−10.68543326780605533228570595311, −9.454040277145368761323210907630, −8.497634992406408054609933682756, −7.88087003866616819454900740600, −6.41462633320973190719177607090, −5.76816064678868981099269830029, −5.37236564966663061184199584862, −3.69978748353934524783639453671, −3.00863541140488744763305444117, −1.61864308978691479339411449070, 1.16151383336220639976469857680, 2.32191895913823019934081750359, 3.76723373041728574813531026528, 4.44578650464405759581634059708, 5.45696137111184569374029820970, 6.65002400127537770754532073802, 7.10660135538313163798099489305, 8.254270154712000934026039978809, 9.787837296836552855714108356506, 10.19933040740318390757984505989

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