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Properties

Label 2-672-168.5-c1-0-27
Degree $2$
Conductor $672$
Sign $-0.848 - 0.529i$
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  + (−0.618 − 1.61i)3-s + (−2.46 − 1.42i)5-s + (1.02 − 2.44i)7-s + (−2.23 + 2.00i)9-s + (−2.42 − 4.20i)11-s + 2.75·13-s + (−0.780 + 4.87i)15-s + (1.75 + 3.03i)17-s + (−3.14 + 5.44i)19-s + (−4.58 − 0.142i)21-s + (−3.15 − 1.82i)23-s + (1.56 + 2.71i)25-s + (4.61 + 2.38i)27-s − 3.90·29-s + (0.858 − 0.495i)31-s + ⋯
L(s)  = 1  + (−0.356 − 0.934i)3-s + (−1.10 − 0.637i)5-s + (0.385 − 0.922i)7-s + (−0.745 + 0.666i)9-s + (−0.731 − 1.26i)11-s + 0.763·13-s + (−0.201 + 1.25i)15-s + (0.425 + 0.736i)17-s + (−0.721 + 1.24i)19-s + (−0.999 − 0.0309i)21-s + (−0.658 − 0.380i)23-s + (0.313 + 0.542i)25-s + (0.888 + 0.458i)27-s − 0.725·29-s + (0.154 − 0.0889i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.848 - 0.529i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.848 - 0.529i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.129884 + 0.453659i\)
\(L(\frac12)\) \(\approx\) \(0.129884 + 0.453659i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.618 + 1.61i)T \)
7 \( 1 + (-1.02 + 2.44i)T \)
good5 \( 1 + (2.46 + 1.42i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.42 + 4.20i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.75T + 13T^{2} \)
17 \( 1 + (-1.75 - 3.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.14 - 5.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.15 + 1.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.90T + 29T^{2} \)
31 \( 1 + (-0.858 + 0.495i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.06 - 0.614i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.10T + 41T^{2} \)
43 \( 1 + 5.11iT - 43T^{2} \)
47 \( 1 + (5.61 - 9.72i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.00 - 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.890 - 0.514i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.24 + 2.15i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.02 - 2.90i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.75iT - 71T^{2} \)
73 \( 1 + (-0.291 + 0.168i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.80 + 4.85i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.138iT - 83T^{2} \)
89 \( 1 + (-0.580 + 1.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.0iT - 97T^{2} \)
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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

−10.39163926191693160202118197817, −8.632739823251536943701642315392, −8.037607380327700394250876804071, −7.73265957731000905177620955645, −6.37119149868040547831971546435, −5.61521538523150300724417039939, −4.32235326424593924306904903071, −3.42313925838864565449589679101, −1.52409291326861942160936890197, −0.26778282190097544423184347893, 2.47193237628440755163408969091, 3.59303063714479241276741597601, 4.62002993083649834195922665432, 5.37570454873483477843447673738, 6.58459314373568467129242961170, 7.58099764248128730559533246837, 8.454581279392354389134055661582, 9.375312514019760807734643937104, 10.22694688728609306228415834406, 11.15564547446629112226221649392

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