↑

Properties

Label 2-552-8.5-c1-0-27
Degree $2$
Conductor $552$
Sign $i$
Analytic cond. $4.40774$
Root an. cond. $2.09946$
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  − 1.41·2-s + i·3-s + 2.00·4-s + 0.585i·5-s − 1.41i·6-s − 2·7-s − 2.82·8-s − 9-s − 0.828i·10-s − 4.24i·11-s + 2.00i·12-s − 1.17i·13-s + 2.82·14-s − 0.585·15-s + 4.00·16-s − 6.82·17-s + ⋯
L(s)  = 1  − 1.00·2-s + 0.577i·3-s + 1.00·4-s + 0.261i·5-s − 0.577i·6-s − 0.755·7-s − 1.00·8-s − 0.333·9-s − 0.261i·10-s − 1.27i·11-s + 0.577i·12-s − 0.324i·13-s + 0.755·14-s − 0.151·15-s + 1.00·16-s − 1.65·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.321991 - 0.321991i\)
\(L(\frac12)\) \(\approx\) \(0.321991 - 0.321991i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 - iT \)
23 \( 1 + T \)
good5 \( 1 - 0.585iT - 5T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + 4.24iT - 11T^{2} \)
13 \( 1 + 1.17iT - 13T^{2} \)
17 \( 1 + 6.82T + 17T^{2} \)
19 \( 1 + 6.24iT - 19T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 - 5.17T + 31T^{2} \)
37 \( 1 - 11.0iT - 37T^{2} \)
41 \( 1 + 8.48T + 41T^{2} \)
43 \( 1 + 10.2iT - 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + 8.58iT - 53T^{2} \)
59 \( 1 + 5.65iT - 59T^{2} \)
61 \( 1 + 12.2iT - 61T^{2} \)
67 \( 1 + 5.07iT - 67T^{2} \)
71 \( 1 + 0.343T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 13.3T + 79T^{2} \)
83 \( 1 - 4.24iT - 83T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + 7.65T + 97T^{2} \)
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

−10.52916408718233688565789150873, −9.660990231706344488123106940727, −8.804645013831591728689365206778, −8.289432882253379858847412934427, −6.76084135437745700697292178869, −6.40755074922469603550163275979, −5.01678744338565166737685041993, −3.42781907914070918814591397223, −2.57656095275045113333994411939, −0.34859101090184599937690069658, 1.58240861346618574264916488950, 2.70455068601222304909108950575, 4.30563068021698946406027746059, 5.88409942047111754674129190272, 6.77020570387366603439982729687, 7.35760878667119291019882245300, 8.472275517593016344573906398387, 9.182779567198765521276119855197, 10.02121843581501888359449926066, 10.81248013966529102676402357296

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