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Properties

Label 2-672-56.27-c1-0-10
Degree $2$
Conductor $672$
Sign $0.713 + 0.700i$
Analytic cond. $5.36594$
Root an. cond. $2.31645$
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  − i·3-s + 1.58·5-s + (2.37 − 1.15i)7-s − 9-s + 2.26·11-s + 0.548·13-s − 1.58i·15-s + 0.433i·17-s + 6.02i·19-s + (−1.15 − 2.37i)21-s − 8.24i·23-s − 2.50·25-s + i·27-s − 0.548i·29-s + 7.50·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.706·5-s + (0.899 − 0.436i)7-s − 0.333·9-s + 0.681·11-s + 0.152·13-s − 0.408i·15-s + 0.105i·17-s + 1.38i·19-s + (−0.252 − 0.519i)21-s − 1.71i·23-s − 0.500·25-s + 0.192i·27-s − 0.101i·29-s + 1.34·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.713 + 0.700i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.713 + 0.700i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75306 - 0.716750i\)
\(L(\frac12)\) \(\approx\) \(1.75306 - 0.716750i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
7 \( 1 + (-2.37 + 1.15i)T \)
good5 \( 1 - 1.58T + 5T^{2} \)
11 \( 1 - 2.26T + 11T^{2} \)
13 \( 1 - 0.548T + 13T^{2} \)
17 \( 1 - 0.433iT - 17T^{2} \)
19 \( 1 - 6.02iT - 19T^{2} \)
23 \( 1 + 8.24iT - 23T^{2} \)
29 \( 1 + 0.548iT - 29T^{2} \)
31 \( 1 - 7.50T + 31T^{2} \)
37 \( 1 + 4.21iT - 37T^{2} \)
41 \( 1 - 7.09iT - 41T^{2} \)
43 \( 1 - 1.82T + 43T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 + 3.71iT - 53T^{2} \)
59 \( 1 + 11.5iT - 59T^{2} \)
61 \( 1 - 12.1T + 61T^{2} \)
67 \( 1 + 9.35T + 67T^{2} \)
71 \( 1 + 1.27iT - 71T^{2} \)
73 \( 1 - 0.867iT - 73T^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 - 6.13iT - 83T^{2} \)
89 \( 1 + 7.95iT - 89T^{2} \)
97 \( 1 - 19.1iT - 97T^{2} \)
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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

−10.36022743259049973301372151614, −9.642339509640792908845054488566, −8.398420156953494987611266846532, −7.976783866518378680938970601442, −6.69730429512381388883631932680, −6.10646526743085691275496221421, −4.94743505288756497301622651562, −3.86120128894253535163784824185, −2.27388327698991915663565718862, −1.23761894764166714743548435675, 1.56141349966314318738156930747, 2.87134981596128243865511007512, 4.23304474297323862300249450602, 5.16409968496921792010113136909, 5.94134248969980748956293966424, 7.05155919211622804092057846629, 8.185443827083553392353087420453, 9.065548638709463017256888782327, 9.625480726535148070773801876173, 10.57763421441802753478298010364

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