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Properties

Label 2-672-56.37-c1-0-4
Degree $2$
Conductor $672$
Sign $0.653 - 0.757i$
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  + (−0.866 + 0.5i)3-s + (1.56 + 0.902i)5-s + (−2.63 + 0.217i)7-s + (0.499 − 0.866i)9-s + (4.48 − 2.58i)11-s − 0.840i·13-s − 1.80·15-s + (2.45 + 4.25i)17-s + (4.87 + 2.81i)19-s + (2.17 − 1.50i)21-s + (−3.05 + 5.28i)23-s + (−0.872 − 1.51i)25-s + 0.999i·27-s − 0.439i·29-s + (3.66 + 6.35i)31-s + ⋯
L(s)  = 1  + (−0.499 + 0.288i)3-s + (0.698 + 0.403i)5-s + (−0.996 + 0.0820i)7-s + (0.166 − 0.288i)9-s + (1.35 − 0.779i)11-s − 0.232i·13-s − 0.465·15-s + (0.595 + 1.03i)17-s + (1.11 + 0.646i)19-s + (0.474 − 0.328i)21-s + (−0.636 + 1.10i)23-s + (−0.174 − 0.302i)25-s + 0.192i·27-s − 0.0816i·29-s + (0.658 + 1.14i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.653 - 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.653 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.653 - 0.757i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.653 - 0.757i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25423 + 0.574629i\)
\(L(\frac12)\) \(\approx\) \(1.25423 + 0.574629i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (2.63 - 0.217i)T \)
good5 \( 1 + (-1.56 - 0.902i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.48 + 2.58i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.840iT - 13T^{2} \)
17 \( 1 + (-2.45 - 4.25i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.87 - 2.81i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.05 - 5.28i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.439iT - 29T^{2} \)
31 \( 1 + (-3.66 - 6.35i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.56 - 2.63i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.23T + 41T^{2} \)
43 \( 1 - 7.34iT - 43T^{2} \)
47 \( 1 + (-2.83 + 4.91i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.15 - 0.669i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.31 + 4.22i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.77 - 2.75i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.647 - 0.373i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.08T + 71T^{2} \)
73 \( 1 + (-3.70 - 6.42i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.68 + 15.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.45iT - 83T^{2} \)
89 \( 1 + (3.10 - 5.37i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.81T + 97T^{2} \)
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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

−10.31751148656564462213643434861, −9.948487173693884509834140755466, −9.159135927396189441071709439933, −8.056462411297245351261682589553, −6.73852250870435659777998249891, −6.12071984002736236981896475437, −5.51324673900912069829978943217, −3.88052308775723995373994241134, −3.16974306192293574921465048509, −1.34566061850793007808747865627, 0.929923943496977117973263782076, 2.40772771072062445103866644405, 3.88384843579438700814075196117, 5.00908812970654164045347436985, 6.00244250923923421864900621971, 6.75411147448118715060565053195, 7.49251354507484566402211492378, 8.976422222045913739660076297829, 9.609924803513254044567427250358, 10.08659301304430572216737445564

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