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Properties

Label 2-672-168.5-c1-0-9
Degree $2$
Conductor $672$
Sign $0.971 + 0.238i$
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  + (−1.09 − 1.34i)3-s + (2.46 + 1.42i)5-s + (1.02 − 2.44i)7-s + (−0.614 + 2.93i)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 + (−4.39 + 1.29i)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  + (−0.630 − 0.776i)3-s + (1.10 + 0.637i)5-s + (0.385 − 0.922i)7-s + (−0.204 + 0.978i)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 + (−0.959 + 0.282i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.238i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.971 + 0.238i$
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: \((2,\ 672,\ (\ :1/2),\ 0.971 + 0.238i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59922 - 0.193085i\)
\(L(\frac12)\) \(\approx\) \(1.59922 - 0.193085i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.09 + 1.34i)T \)
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

−10.42295093250994558505470524328, −9.947844467730981768246264147175, −8.711853319147636049179021986247, −7.52855203974511712058677032585, −6.82335431144988845471950835801, −6.24539126802008088393045265743, −5.15185893522737674821687007350, −4.02207745046063650815403115243, −2.27496749400272544180108787600, −1.34477211286664604533231132912, 1.18373854664409466979465375068, 2.80593717430398578178966210624, 4.25483869003077469244109569705, 5.15359319566322050535920167842, 6.09036166589536450998168106612, 6.37178540977287307563636255565, 8.452348085568079853240481462890, 8.946435526663240816551619918886, 9.452019443024099240630797576696, 10.81609491757944396788211192103

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