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Properties

Label 2-672-12.11-c3-0-12
Degree $2$
Conductor $672$
Sign $0.755 - 0.655i$
Analytic cond. $39.6492$
Root an. cond. $6.29676$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.18 − 0.369i)3-s − 10.4i·5-s − 7i·7-s + (26.7 + 3.82i)9-s − 53.7·11-s − 40.4·13-s + (−3.85 + 54.0i)15-s − 23.2i·17-s + 138. i·19-s + (−2.58 + 36.2i)21-s − 23.4·23-s + 16.3·25-s + (−137. − 29.7i)27-s − 19.7i·29-s − 251. i·31-s + ⋯
L(s)  = 1  + (−0.997 − 0.0710i)3-s − 0.932i·5-s − 0.377i·7-s + (0.989 + 0.141i)9-s − 1.47·11-s − 0.863·13-s + (−0.0662 + 0.929i)15-s − 0.331i·17-s + 1.67i·19-s + (−0.0268 + 0.377i)21-s − 0.212·23-s + 0.131·25-s + (−0.977 − 0.211i)27-s − 0.126i·29-s − 1.45i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.755 - 0.655i$
Analytic conductor: \(39.6492\)
Root analytic conductor: \(6.29676\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :3/2),\ 0.755 - 0.655i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6265163138\)
\(L(\frac12)\) \(\approx\) \(0.6265163138\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.18 + 0.369i)T \)
7 \( 1 + 7iT \)
good5 \( 1 + 10.4iT - 125T^{2} \)
11 \( 1 + 53.7T + 1.33e3T^{2} \)
13 \( 1 + 40.4T + 2.19e3T^{2} \)
17 \( 1 + 23.2iT - 4.91e3T^{2} \)
19 \( 1 - 138. iT - 6.85e3T^{2} \)
23 \( 1 + 23.4T + 1.21e4T^{2} \)
29 \( 1 + 19.7iT - 2.43e4T^{2} \)
31 \( 1 + 251. iT - 2.97e4T^{2} \)
37 \( 1 + 123.T + 5.06e4T^{2} \)
41 \( 1 + 45.1iT - 6.89e4T^{2} \)
43 \( 1 - 229. iT - 7.95e4T^{2} \)
47 \( 1 + 559.T + 1.03e5T^{2} \)
53 \( 1 + 289. iT - 1.48e5T^{2} \)
59 \( 1 - 371.T + 2.05e5T^{2} \)
61 \( 1 - 518.T + 2.26e5T^{2} \)
67 \( 1 - 668. iT - 3.00e5T^{2} \)
71 \( 1 - 1.05e3T + 3.57e5T^{2} \)
73 \( 1 - 1.16e3T + 3.89e5T^{2} \)
79 \( 1 - 1.09e3iT - 4.93e5T^{2} \)
83 \( 1 - 253.T + 5.71e5T^{2} \)
89 \( 1 + 190. iT - 7.04e5T^{2} \)
97 \( 1 - 117.T + 9.12e5T^{2} \)
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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

−10.05355725491909946476582412596, −9.760784173391722498484098850971, −8.194007630400908444891598409435, −7.69887494689800973264351313576, −6.57347661614619610211734726337, −5.40759823482876448765541052637, −5.01441925978733600937190126614, −3.89661311595593734302262344303, −2.15252431639340111487241101460, −0.76280761474436574048180857051, 0.28821646246350180739342414663, 2.20485476562703141617968728697, 3.22159303469174588467680363519, 4.85851143979530614806473173287, 5.30025700311977041128652757677, 6.59685009550814820435305916689, 7.07827904449493639387061939826, 8.117091773660401631637158935370, 9.369842096876484066211384685223, 10.33889839356324860326884938062

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