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Properties

Label 2-714-17.16-c1-0-12
Degree $2$
Conductor $714$
Sign $0.575 + 0.817i$
Analytic cond. $5.70131$
Root an. cond. $2.38774$
Motivic weight $1$
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 + 1.13i·5-s − i·6-s − i·7-s − 8-s − 9-s − 1.13i·10-s − 6.43i·11-s + i·12-s + 0.590·13-s + i·14-s − 1.13·15-s + 16-s + (−2.37 − 3.37i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577i·3-s + 0.5·4-s + 0.509i·5-s − 0.408i·6-s − 0.377i·7-s − 0.353·8-s − 0.333·9-s − 0.360i·10-s − 1.93i·11-s + 0.288i·12-s + 0.163·13-s + 0.267i·14-s − 0.294·15-s + 0.250·16-s + (−0.575 − 0.817i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 + 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.575 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(714\)    =    \(2 \cdot 3 \cdot 7 \cdot 17\)
Sign: $0.575 + 0.817i$
Analytic conductor: \(5.70131\)
Root analytic conductor: \(2.38774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{714} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 714,\ (\ :1/2),\ 0.575 + 0.817i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.758723 - 0.393914i\)
\(L(\frac12)\) \(\approx\) \(0.758723 - 0.393914i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - iT \)
7 \( 1 + iT \)
17 \( 1 + (2.37 + 3.37i)T \)
good5 \( 1 - 1.13iT - 5T^{2} \)
11 \( 1 + 6.43iT - 11T^{2} \)
13 \( 1 - 0.590T + 13T^{2} \)
19 \( 1 + 2.54T + 19T^{2} \)
23 \( 1 + 6.74iT - 23T^{2} \)
29 \( 1 - 0.270iT - 29T^{2} \)
31 \( 1 - 2.27iT - 31T^{2} \)
37 \( 1 + 2.59iT - 37T^{2} \)
41 \( 1 - 3.84iT - 41T^{2} \)
43 \( 1 - 7.88T + 43T^{2} \)
47 \( 1 + 2.27T + 47T^{2} \)
53 \( 1 - 1.88T + 53T^{2} \)
59 \( 1 + 9.29T + 59T^{2} \)
61 \( 1 + 7.84iT - 61T^{2} \)
67 \( 1 - 7.88T + 67T^{2} \)
71 \( 1 + 9.56iT - 71T^{2} \)
73 \( 1 + 0.0422iT - 73T^{2} \)
79 \( 1 + 6.16iT - 79T^{2} \)
83 \( 1 - 15.4T + 83T^{2} \)
89 \( 1 - 5.41T + 89T^{2} \)
97 \( 1 + 2.23iT - 97T^{2} \)
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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

−10.61727761405350151813577914403, −9.306794517864447449368679994952, −8.705871994899865900543946412080, −7.918212682663633734977095075271, −6.70118831671290766210000686232, −6.09472359310616724416847038462, −4.79402840828775678681483371820, −3.50938324052936016161722994037, −2.62382886094295225463874216811, −0.57455284111497763863962920133, 1.49260524531326840481799030863, 2.39131791069643734982467531466, 4.10658307457549129703875040898, 5.24868418824821290200748940601, 6.36867458553087968361974993280, 7.20349123347583625417468114764, 7.962431141955610705398229662214, 8.887744831050872827973123678898, 9.523838798588833664133183502560, 10.44410357938392488670906431859

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