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Properties

Label 2-672-224.187-c1-0-36
Degree $2$
Conductor $672$
Sign $0.910 + 0.413i$
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  + (1.31 − 0.523i)2-s + (−0.608 + 0.793i)3-s + (1.45 − 1.37i)4-s + (0.309 + 0.403i)5-s + (−0.384 + 1.36i)6-s + (2.60 + 0.434i)7-s + (1.18 − 2.56i)8-s + (−0.258 − 0.965i)9-s + (0.617 + 0.367i)10-s + (−0.333 − 2.53i)11-s + (0.206 + 1.98i)12-s + (−0.888 − 2.14i)13-s + (3.65 − 0.794i)14-s − 0.508·15-s + (0.219 − 3.99i)16-s + (3.23 + 5.60i)17-s + ⋯
L(s)  = 1  + (0.929 − 0.369i)2-s + (−0.351 + 0.458i)3-s + (0.726 − 0.687i)4-s + (0.138 + 0.180i)5-s + (−0.157 + 0.555i)6-s + (0.986 + 0.164i)7-s + (0.420 − 0.907i)8-s + (−0.0862 − 0.321i)9-s + (0.195 + 0.116i)10-s + (−0.100 − 0.763i)11-s + (0.0596 + 0.574i)12-s + (−0.246 − 0.594i)13-s + (0.977 − 0.212i)14-s − 0.131·15-s + (0.0549 − 0.998i)16-s + (0.784 + 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.910 + 0.413i$
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: \((2,\ 672,\ (\ :1/2),\ 0.910 + 0.413i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.60348 - 0.562754i\)
\(L(\frac12)\) \(\approx\) \(2.60348 - 0.562754i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.31 + 0.523i)T \)
3 \( 1 + (0.608 - 0.793i)T \)
7 \( 1 + (-2.60 - 0.434i)T \)
good5 \( 1 + (-0.309 - 0.403i)T + (-1.29 + 4.82i)T^{2} \)
11 \( 1 + (0.333 + 2.53i)T + (-10.6 + 2.84i)T^{2} \)
13 \( 1 + (0.888 + 2.14i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + (-3.23 - 5.60i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.249 - 1.89i)T + (-18.3 - 4.91i)T^{2} \)
23 \( 1 + (-1.54 + 0.414i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-3.25 - 7.85i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (0.881 + 1.52i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.125 + 0.0962i)T + (9.57 - 35.7i)T^{2} \)
41 \( 1 + (0.144 + 0.144i)T + 41iT^{2} \)
43 \( 1 + (3.89 + 1.61i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + (1.81 + 1.05i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.29 - 0.433i)T + (51.1 - 13.7i)T^{2} \)
59 \( 1 + (0.641 + 4.87i)T + (-56.9 + 15.2i)T^{2} \)
61 \( 1 + (-0.103 + 0.786i)T + (-58.9 - 15.7i)T^{2} \)
67 \( 1 + (10.8 + 8.33i)T + (17.3 + 64.7i)T^{2} \)
71 \( 1 + (8.96 - 8.96i)T - 71iT^{2} \)
73 \( 1 + (1.58 - 5.90i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.46 - 2.53i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.62 - 3.98i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-2.20 - 8.23i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 - 9.62iT - 97T^{2} \)
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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

−10.59073165448333452823356004189, −10.09943733112314968021827116387, −8.688976801172394274356899170391, −7.82419591289634552571774102256, −6.52450520178531920039679727844, −5.66459523940521095611109484175, −5.01335106518179585901127760449, −3.92446416808869370415325937293, −2.90592604344357555909840232900, −1.40101453556568107898604657356, 1.61888339244696079481167090646, 2.85408367728248032536423818872, 4.49628572349499510222495638065, 4.96999904362557418171420226540, 5.95851413021532313965368324999, 7.19971106401382268940503178127, 7.44993249125710568816050318564, 8.585859193376055948215211687617, 9.754196965748766250588118405237, 10.91318084225277431426061736656

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