Properties

Label 2-672-28.27-c1-0-8
Degree $2$
Conductor $672$
Sign $0.993 + 0.110i$
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  − 3-s + 0.922i·5-s + (2.06 − 1.65i)7-s + 9-s − 1.61i·11-s + 2.82i·13-s − 0.922i·15-s + 0.922i·17-s + 0.540·19-s + (−2.06 + 1.65i)21-s − 7.05i·23-s + 4.14·25-s − 27-s + 1.01·29-s + 8.67·31-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.412i·5-s + (0.781 − 0.624i)7-s + 0.333·9-s − 0.487i·11-s + 0.784i·13-s − 0.238i·15-s + 0.223i·17-s + 0.123·19-s + (−0.450 + 0.360i)21-s − 1.47i·23-s + 0.829·25-s − 0.192·27-s + 0.188·29-s + 1.55·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.993 + 0.110i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.993 + 0.110i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37581 - 0.0763885i\)
\(L(\frac12)\) \(\approx\) \(1.37581 - 0.0763885i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + (-2.06 + 1.65i)T \)
good5 \( 1 - 0.922iT - 5T^{2} \)
11 \( 1 + 1.61iT - 11T^{2} \)
13 \( 1 - 2.82iT - 13T^{2} \)
17 \( 1 - 0.922iT - 17T^{2} \)
19 \( 1 - 0.540T + 19T^{2} \)
23 \( 1 + 7.05iT - 23T^{2} \)
29 \( 1 - 1.01T + 29T^{2} \)
31 \( 1 - 8.67T + 31T^{2} \)
37 \( 1 - 8.19T + 37T^{2} \)
41 \( 1 - 6.57iT - 41T^{2} \)
43 \( 1 - 4.96iT - 43T^{2} \)
47 \( 1 - 2.60T + 47T^{2} \)
53 \( 1 - 5.59T + 53T^{2} \)
59 \( 1 - 1.65T + 59T^{2} \)
61 \( 1 + 7.01iT - 61T^{2} \)
67 \( 1 - 6.80iT - 67T^{2} \)
71 \( 1 + 5.21iT - 71T^{2} \)
73 \( 1 + 7.50iT - 73T^{2} \)
79 \( 1 + 7.11iT - 79T^{2} \)
83 \( 1 + 17.9T + 83T^{2} \)
89 \( 1 - 17.9iT - 89T^{2} \)
97 \( 1 + 4.18iT - 97T^{2} \)
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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

−10.63718805955167987703286832360, −9.857994986560418059227999909767, −8.648972287972147992963750482344, −7.86522456721895799892205411210, −6.78806197549801688663838293581, −6.19858511858803703957232927877, −4.84982693620670719116515227344, −4.19377472299050687178511805027, −2.67802252048264656560045916424, −1.06123788719236089520345747959, 1.16556609990206262074365838610, 2.64542618365220786862307016810, 4.22902707303103262258035318695, 5.18911218524854693542817094170, 5.76121692051386185907343282082, 7.04017446435037993483323890131, 7.919891378173166098451520783719, 8.768970004986895051829494395465, 9.716017398920146445468816061661, 10.55735835879705620456582384503

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