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Properties

Label 2-672-224.187-c1-0-20
Degree $2$
Conductor $672$
Sign $-0.934 - 0.356i$
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  + (−0.0456 + 1.41i)2-s + (−0.608 + 0.793i)3-s + (−1.99 − 0.129i)4-s + (2.11 + 2.75i)5-s + (−1.09 − 0.896i)6-s + (2.45 − 0.984i)7-s + (0.273 − 2.81i)8-s + (−0.258 − 0.965i)9-s + (−3.98 + 2.86i)10-s + (0.570 + 4.33i)11-s + (1.31 − 1.50i)12-s + (0.414 + 1.00i)13-s + (1.27 + 3.51i)14-s − 3.47·15-s + (3.96 + 0.515i)16-s + (1.39 + 2.41i)17-s + ⋯
L(s)  = 1  + (−0.0322 + 0.999i)2-s + (−0.351 + 0.458i)3-s + (−0.997 − 0.0645i)4-s + (0.945 + 1.23i)5-s + (−0.446 − 0.366i)6-s + (0.928 − 0.372i)7-s + (0.0966 − 0.995i)8-s + (−0.0862 − 0.321i)9-s + (−1.26 + 0.904i)10-s + (0.171 + 1.30i)11-s + (0.380 − 0.434i)12-s + (0.114 + 0.277i)13-s + (0.342 + 0.939i)14-s − 0.896·15-s + (0.991 + 0.128i)16-s + (0.337 + 0.584i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.934 - 0.356i$
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: \((2,\ 672,\ (\ :1/2),\ -0.934 - 0.356i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.272361 + 1.47597i\)
\(L(\frac12)\) \(\approx\) \(0.272361 + 1.47597i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0456 - 1.41i)T \)
3 \( 1 + (0.608 - 0.793i)T \)
7 \( 1 + (-2.45 + 0.984i)T \)
good5 \( 1 + (-2.11 - 2.75i)T + (-1.29 + 4.82i)T^{2} \)
11 \( 1 + (-0.570 - 4.33i)T + (-10.6 + 2.84i)T^{2} \)
13 \( 1 + (-0.414 - 1.00i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + (-1.39 - 2.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.148 + 1.12i)T + (-18.3 - 4.91i)T^{2} \)
23 \( 1 + (-2.58 + 0.693i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.73 - 4.18i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (4.27 + 7.39i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.587 + 0.450i)T + (9.57 - 35.7i)T^{2} \)
41 \( 1 + (1.90 + 1.90i)T + 41iT^{2} \)
43 \( 1 + (8.53 + 3.53i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + (10.4 + 6.03i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.35 - 0.441i)T + (51.1 - 13.7i)T^{2} \)
59 \( 1 + (0.135 + 1.02i)T + (-56.9 + 15.2i)T^{2} \)
61 \( 1 + (0.552 - 4.19i)T + (-58.9 - 15.7i)T^{2} \)
67 \( 1 + (-7.44 - 5.71i)T + (17.3 + 64.7i)T^{2} \)
71 \( 1 + (-11.3 + 11.3i)T - 71iT^{2} \)
73 \( 1 + (1.05 - 3.92i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.309 + 0.535i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.13 - 0.883i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-3.18 - 11.9i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 - 13.4iT - 97T^{2} \)
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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

−10.58712926090805117072960722894, −10.00548294453392808380782541155, −9.289391573710769955383409853191, −8.122370256131640316891715281631, −7.06488587724476545295581560457, −6.63322391976047276109881524400, −5.51866333310869420811999456853, −4.73950225389767425301912884751, −3.63157555113589292851111617025, −1.85582795040538835952070934955, 0.951019434794459852824630784465, 1.78247679113744480046782191357, 3.19775961177826284435416589950, 4.85337103525512600539037391999, 5.27120394941620277249974603647, 6.18559772861190482126520880553, 7.970303991383071813957182824394, 8.545424354245804079897416377446, 9.260469257978859544041501497261, 10.17225382360072848567843693462

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