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Properties

Label 2-672-168.107-c1-0-25
Degree $2$
Conductor $672$
Sign $-0.0454 + 0.998i$
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  + (0.987 − 1.42i)3-s + (1.77 − 3.07i)5-s + (−0.793 + 2.52i)7-s + (−1.04 − 2.81i)9-s + (0.396 − 0.229i)11-s + 0.799i·13-s + (−2.62 − 5.57i)15-s + (5.48 − 3.16i)17-s + (2.61 − 4.53i)19-s + (2.80 + 3.62i)21-s + (−1.55 + 2.68i)23-s + (−3.82 − 6.61i)25-s + (−5.03 − 1.28i)27-s − 2.60·29-s + (−4.42 + 2.55i)31-s + ⋯
L(s)  = 1  + (0.570 − 0.821i)3-s + (0.795 − 1.37i)5-s + (−0.299 + 0.953i)7-s + (−0.349 − 0.936i)9-s + (0.119 − 0.0690i)11-s + 0.221i·13-s + (−0.678 − 1.43i)15-s + (1.33 − 0.767i)17-s + (0.600 − 1.03i)19-s + (0.612 + 0.790i)21-s + (−0.323 + 0.559i)23-s + (−0.764 − 1.32i)25-s + (−0.969 − 0.246i)27-s − 0.483·29-s + (−0.794 + 0.458i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0454 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0454 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.0454 + 0.998i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.0454 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36806 - 1.43175i\)
\(L(\frac12)\) \(\approx\) \(1.36806 - 1.43175i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.987 + 1.42i)T \)
7 \( 1 + (0.793 - 2.52i)T \)
good5 \( 1 + (-1.77 + 3.07i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.396 + 0.229i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.799iT - 13T^{2} \)
17 \( 1 + (-5.48 + 3.16i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.61 + 4.53i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.55 - 2.68i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.60T + 29T^{2} \)
31 \( 1 + (4.42 - 2.55i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.89 + 1.67i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.1iT - 41T^{2} \)
43 \( 1 + 3.56T + 43T^{2} \)
47 \( 1 + (-2.15 + 3.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.16 - 2.02i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.49 + 0.864i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.60 - 2.66i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.00979 + 0.0169i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.04T + 71T^{2} \)
73 \( 1 + (-4.15 - 7.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-10.8 - 6.28i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.694iT - 83T^{2} \)
89 \( 1 + (-7.02 - 4.05i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.05T + 97T^{2} \)
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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

−9.731074623525236474319698040614, −9.352741255333461294727981671632, −8.686856249447476017613409443419, −7.81078589898530498210631421721, −6.73811668030010362790073119906, −5.63786084997223137936134320262, −5.11780887158178350736269835648, −3.40028989636158812846409190320, −2.21132700434993000056065984500, −1.07043757071507766182307170335, 1.99217638538576373870997584096, 3.35019886992347232424303298698, 3.77951459014214125908812035734, 5.37324899490790599022827867599, 6.22862948685174109395008411183, 7.32981516720244522751962434966, 7.994998442160396849806893306568, 9.318074357632843878651825324050, 10.21783214801160026401356972273, 10.27873003985308187772340585951

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