Properties

Label 2-672-8.5-c1-0-4
Degree $2$
Conductor $672$
Sign $0.806 - 0.591i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  i·3-s + 0.467i·5-s − 7-s − 9-s + 4.87i·11-s + 4.56i·13-s + 0.467·15-s + 6.09·17-s + 1.34i·19-s + i·21-s + 4.09·23-s + 4.78·25-s + i·27-s − 7.78i·29-s − 4.40·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.208i·5-s − 0.377·7-s − 0.333·9-s + 1.47i·11-s + 1.26i·13-s + 0.120·15-s + 1.47·17-s + 0.308i·19-s + 0.218i·21-s + 0.854·23-s + 0.956·25-s + 0.192i·27-s − 1.44i·29-s − 0.791·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.806 - 0.591i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.806 - 0.591i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31056 + 0.428807i\)
\(L(\frac12)\) \(\approx\) \(1.31056 + 0.428807i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
7 \( 1 + T \)
good5 \( 1 - 0.467iT - 5T^{2} \)
11 \( 1 - 4.87iT - 11T^{2} \)
13 \( 1 - 4.56iT - 13T^{2} \)
17 \( 1 - 6.09T + 17T^{2} \)
19 \( 1 - 1.34iT - 19T^{2} \)
23 \( 1 - 4.09T + 23T^{2} \)
29 \( 1 + 7.78iT - 29T^{2} \)
31 \( 1 + 4.40T + 31T^{2} \)
37 \( 1 - 4.40iT - 37T^{2} \)
41 \( 1 + 6.09T + 41T^{2} \)
43 \( 1 - 4.15iT - 43T^{2} \)
47 \( 1 - 6.68T + 47T^{2} \)
53 \( 1 - 1.34iT - 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 - 5.49iT - 61T^{2} \)
67 \( 1 - 5.90iT - 67T^{2} \)
71 \( 1 - 4.72T + 71T^{2} \)
73 \( 1 + 12.0T + 73T^{2} \)
79 \( 1 - 16.1T + 79T^{2} \)
83 \( 1 + 13.7iT - 83T^{2} \)
89 \( 1 - 7.96T + 89T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.47834151472936418585612906344, −9.723643523347426924328954326803, −8.960316342805901739943699759167, −7.74162602681861297069128320253, −7.09413294264565331699204308731, −6.34235846696201103877821376626, −5.15895956576428974340053342752, −4.05164891612955381973817877322, −2.73088562185847073874759392351, −1.49236221692041960328235589386, 0.813460356922776492031875778548, 3.06162852962262700032813783632, 3.53302156970907584264258685596, 5.22075933387531208908932175894, 5.58034990301373412072784717664, 6.85672637921539069142190871546, 7.987331190867382619397088736986, 8.758637795991941172371417518774, 9.504723442987829945549337758234, 10.74644787661730475328474881588

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