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Properties

Label 2-672-168.5-c1-0-7
Degree $2$
Conductor $672$
Sign $0.999 + 0.00446i$
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  + (0.0574 − 1.73i)3-s + (0.461 + 0.266i)5-s + (−0.489 + 2.60i)7-s + (−2.99 − 0.198i)9-s + (2.28 + 3.96i)11-s + 4.97·13-s + (0.487 − 0.783i)15-s + (2.16 + 3.74i)17-s + (0.921 − 1.59i)19-s + (4.47 + 0.996i)21-s + (0.103 + 0.0596i)23-s + (−2.35 − 4.08i)25-s + (−0.516 + 5.17i)27-s + 7.74·29-s + (1.93 − 1.11i)31-s + ⋯
L(s)  = 1  + (0.0331 − 0.999i)3-s + (0.206 + 0.119i)5-s + (−0.184 + 0.982i)7-s + (−0.997 − 0.0663i)9-s + (0.689 + 1.19i)11-s + 1.38·13-s + (0.125 − 0.202i)15-s + (0.524 + 0.907i)17-s + (0.211 − 0.366i)19-s + (0.976 + 0.217i)21-s + (0.0215 + 0.0124i)23-s + (−0.471 − 0.816i)25-s + (−0.0993 + 0.995i)27-s + 1.43·29-s + (0.347 − 0.200i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.999 + 0.00446i$
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: \((2,\ 672,\ (\ :1/2),\ 0.999 + 0.00446i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61434 - 0.00360123i\)
\(L(\frac12)\) \(\approx\) \(1.61434 - 0.00360123i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0574 + 1.73i)T \)
7 \( 1 + (0.489 - 2.60i)T \)
good5 \( 1 + (-0.461 - 0.266i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.28 - 3.96i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.97T + 13T^{2} \)
17 \( 1 + (-2.16 - 3.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.921 + 1.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.103 - 0.0596i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.74T + 29T^{2} \)
31 \( 1 + (-1.93 + 1.11i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.02 + 4.05i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.60T + 41T^{2} \)
43 \( 1 + 1.87iT - 43T^{2} \)
47 \( 1 + (2.91 - 5.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.29 - 3.97i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.71 + 3.29i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.07 - 1.86i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.4 + 6.05i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.20iT - 71T^{2} \)
73 \( 1 + (-8.35 + 4.82i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.0228 + 0.0396i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.86iT - 83T^{2} \)
89 \( 1 + (8.23 - 14.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.18iT - 97T^{2} \)
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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

−10.53715368538741107898027308701, −9.511456315810753426233396838900, −8.629631211665419551532660561054, −8.014104764256442845754058308822, −6.71122926723231481245851134097, −6.29424954063561118106480124263, −5.28837309654169321810537483585, −3.79362895379232074543362663343, −2.48379503768180786772031951082, −1.43243774524515035996001337884, 1.02594987004240991668615464048, 3.27139251181619838678244380150, 3.74193779237706083651244797368, 4.97135657441810674518989994167, 5.94213733846956209526949997895, 6.81895233635193483864013265170, 8.215110692847137256349925228051, 8.781851037163805309432133220719, 9.797200388757859585775333551812, 10.38844618021952274137056751296

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