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Properties

Label 2-688-43.42-c4-0-0
Degree $2$
Conductor $688$
Sign $0.731 - 0.681i$
Analytic cond. $71.1185$
Root an. cond. $8.43318$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 14.5i·3-s + 9.40i·5-s − 53.5i·7-s − 129.·9-s − 151.·11-s − 319.·13-s + 136.·15-s + 32.2·17-s − 304. i·19-s − 777.·21-s + 199.·23-s + 536.·25-s + 708. i·27-s + 268. i·29-s − 665.·31-s + ⋯
L(s)  = 1  − 1.61i·3-s + 0.376i·5-s − 1.09i·7-s − 1.60·9-s − 1.24·11-s − 1.89·13-s + 0.606·15-s + 0.111·17-s − 0.842i·19-s − 1.76·21-s + 0.376·23-s + 0.858·25-s + 0.971i·27-s + 0.319i·29-s − 0.692·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 - 0.681i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(688\)    =    \(2^{4} \cdot 43\)
Sign: $0.731 - 0.681i$
Analytic conductor: \(71.1185\)
Root analytic conductor: \(8.43318\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{688} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 688,\ (\ :2),\ 0.731 - 0.681i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.09623229931\)
\(L(\frac12)\) \(\approx\) \(0.09623229931\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 + (1.35e3 - 1.26e3i)T \)
good3 \( 1 + 14.5iT - 81T^{2} \)
5 \( 1 - 9.40iT - 625T^{2} \)
7 \( 1 + 53.5iT - 2.40e3T^{2} \)
11 \( 1 + 151.T + 1.46e4T^{2} \)
13 \( 1 + 319.T + 2.85e4T^{2} \)
17 \( 1 - 32.2T + 8.35e4T^{2} \)
19 \( 1 + 304. iT - 1.30e5T^{2} \)
23 \( 1 - 199.T + 2.79e5T^{2} \)
29 \( 1 - 268. iT - 7.07e5T^{2} \)
31 \( 1 + 665.T + 9.23e5T^{2} \)
37 \( 1 + 1.17e3iT - 1.87e6T^{2} \)
41 \( 1 - 212.T + 2.82e6T^{2} \)
47 \( 1 - 3.10e3T + 4.87e6T^{2} \)
53 \( 1 - 2.66e3T + 7.89e6T^{2} \)
59 \( 1 + 2.74e3T + 1.21e7T^{2} \)
61 \( 1 - 5.88e3iT - 1.38e7T^{2} \)
67 \( 1 - 1.09e3T + 2.01e7T^{2} \)
71 \( 1 + 5.84e3iT - 2.54e7T^{2} \)
73 \( 1 + 663. iT - 2.83e7T^{2} \)
79 \( 1 + 6.32e3T + 3.89e7T^{2} \)
83 \( 1 + 8.35e3T + 4.74e7T^{2} \)
89 \( 1 - 4.25e3iT - 6.27e7T^{2} \)
97 \( 1 + 3.58e3T + 8.85e7T^{2} \)
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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

−10.20309238009551724822432353761, −8.978505050957719326093632340533, −7.78751381202732189998674329341, −7.23476427500786138524980884150, −6.97923509610095736472630968446, −5.61648397205131356556999485319, −4.63464235064594332120334946533, −2.94011652873405604976782059912, −2.24757914800371568924526152474, −0.846594512369610318939104590746, 0.02775135333818372909710693053, 2.31637729017239491785055490373, 3.13478004916774623269403602903, 4.47706861857237777874168454057, 5.18956838568242411940237698372, 5.64036905548249312795264570998, 7.28883428872624476701161304109, 8.364053485980226017010759153130, 9.062611623500231324444709492460, 9.912984445653453410381191511448

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