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Properties

Label 2-672-28.27-c1-0-10
Degree $2$
Conductor $672$
Sign $0.645 + 0.763i$
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 − 2.31i·5-s + (−0.222 + 2.63i)7-s + 9-s − 3.58i·11-s − 2.82i·13-s − 2.31i·15-s − 2.31i·17-s + 7.90·19-s + (−0.222 + 2.63i)21-s + 0.130i·23-s − 0.355·25-s + 27-s + 0.199·29-s + 3.45·31-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.03i·5-s + (−0.0839 + 0.996i)7-s + 0.333·9-s − 1.08i·11-s − 0.784i·13-s − 0.597i·15-s − 0.561i·17-s + 1.81·19-s + (−0.0484 + 0.575i)21-s + 0.0271i·23-s − 0.0711·25-s + 0.192·27-s + 0.0371·29-s + 0.620·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 + 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.645 + 0.763i$
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: \((2,\ 672,\ (\ :1/2),\ 0.645 + 0.763i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65524 - 0.768642i\)
\(L(\frac12)\) \(\approx\) \(1.65524 - 0.768642i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.222 - 2.63i)T \)
good5 \( 1 + 2.31iT - 5T^{2} \)
11 \( 1 + 3.58iT - 11T^{2} \)
13 \( 1 + 2.82iT - 13T^{2} \)
17 \( 1 + 2.31iT - 17T^{2} \)
19 \( 1 - 7.90T + 19T^{2} \)
23 \( 1 - 0.130iT - 23T^{2} \)
29 \( 1 - 0.199T + 29T^{2} \)
31 \( 1 - 3.45T + 31T^{2} \)
37 \( 1 + 11.5T + 37T^{2} \)
41 \( 1 + 7.97iT - 41T^{2} \)
43 \( 1 - 4.38iT - 43T^{2} \)
47 \( 1 + 6.54T + 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 - 9.65T + 59T^{2} \)
61 \( 1 + 6.19iT - 61T^{2} \)
67 \( 1 - 9.01iT - 67T^{2} \)
71 \( 1 - 4.75iT - 71T^{2} \)
73 \( 1 - 10.2iT - 73T^{2} \)
79 \( 1 - 4.24iT - 79T^{2} \)
83 \( 1 + 0.768T + 83T^{2} \)
89 \( 1 - 17.7iT - 89T^{2} \)
97 \( 1 + 9.02iT - 97T^{2} \)
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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

−10.15786154477212626421050040497, −9.328465975615740466338061838940, −8.625568927720768901096864963061, −8.124262889778495365096614753594, −6.94154224850415731897684589678, −5.48760588677091343740978368535, −5.21386294053278868920723888671, −3.59624359834717057980880729075, −2.68011911303121156072842646291, −1.02397296848201914162897612594, 1.65803099793457107463985576335, 3.03320621859166994473393779587, 3.90313934560955782230892078976, 4.98620253082534370449775467490, 6.56963969811064549733775148146, 7.12431829489518217298349550500, 7.77001959656079473398634403547, 8.993532318664787924399309950792, 10.01625076533030172912006852641, 10.30598123158657591587698703594

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