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Properties

Label 2-672-224.187-c1-0-53
Degree $2$
Conductor $672$
Sign $0.318 + 0.947i$
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  + (0.807 + 1.16i)2-s + (0.608 − 0.793i)3-s + (−0.696 + 1.87i)4-s + (−1.16 − 1.51i)5-s + (1.41 + 0.0664i)6-s + (−2.24 − 1.39i)7-s + (−2.73 + 0.704i)8-s + (−0.258 − 0.965i)9-s + (0.822 − 2.57i)10-s + (−0.685 − 5.21i)11-s + (1.06 + 1.69i)12-s + (0.437 + 1.05i)13-s + (−0.186 − 3.73i)14-s − 1.91·15-s + (−3.02 − 2.61i)16-s + (0.928 + 1.60i)17-s + ⋯
L(s)  = 1  + (0.570 + 0.821i)2-s + (0.351 − 0.458i)3-s + (−0.348 + 0.937i)4-s + (−0.520 − 0.678i)5-s + (0.576 + 0.0271i)6-s + (−0.848 − 0.529i)7-s + (−0.968 + 0.248i)8-s + (−0.0862 − 0.321i)9-s + (0.260 − 0.815i)10-s + (−0.206 − 1.57i)11-s + (0.306 + 0.489i)12-s + (0.121 + 0.292i)13-s + (−0.0498 − 0.998i)14-s − 0.494·15-s + (−0.757 − 0.653i)16-s + (0.225 + 0.389i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.318 + 0.947i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.318 + 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03433 - 0.743635i\)
\(L(\frac12)\) \(\approx\) \(1.03433 - 0.743635i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.807 - 1.16i)T \)
3 \( 1 + (-0.608 + 0.793i)T \)
7 \( 1 + (2.24 + 1.39i)T \)
good5 \( 1 + (1.16 + 1.51i)T + (-1.29 + 4.82i)T^{2} \)
11 \( 1 + (0.685 + 5.21i)T + (-10.6 + 2.84i)T^{2} \)
13 \( 1 + (-0.437 - 1.05i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + (-0.928 - 1.60i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.853 + 6.48i)T + (-18.3 - 4.91i)T^{2} \)
23 \( 1 + (1.60 - 0.431i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.11 + 2.70i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-4.29 - 7.43i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.79 - 2.14i)T + (9.57 - 35.7i)T^{2} \)
41 \( 1 + (5.87 + 5.87i)T + 41iT^{2} \)
43 \( 1 + (1.64 + 0.682i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + (0.0886 + 0.0511i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.81 + 0.633i)T + (51.1 - 13.7i)T^{2} \)
59 \( 1 + (-0.411 - 3.12i)T + (-56.9 + 15.2i)T^{2} \)
61 \( 1 + (0.957 - 7.27i)T + (-58.9 - 15.7i)T^{2} \)
67 \( 1 + (8.02 + 6.16i)T + (17.3 + 64.7i)T^{2} \)
71 \( 1 + (-6.41 + 6.41i)T - 71iT^{2} \)
73 \( 1 + (4.24 - 15.8i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.92 + 10.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.78 + 2.81i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (3.73 + 13.9i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 - 7.23iT - 97T^{2} \)
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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

−10.28525822564256750788551590824, −8.845860201742849606494980907409, −8.652482638824580959641652663533, −7.61567548959787907266997820691, −6.76040925526671171885734145729, −5.99515450407399057369538401516, −4.85321398174737989533947015445, −3.74495606467178044619093335828, −2.96869192558349898401176195078, −0.52629230246826582200221885815, 2.07283529794228068438710744608, 3.14005584120742891998119571912, 3.87432893593762706745290114513, 4.98203500348942161022537275066, 6.01796728756008610118199898710, 7.08478764232687841100371985301, 8.145300962899549822389787882016, 9.444097423341984088080715360539, 9.909066353382604935857263608018, 10.54283488941183389432404897150

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