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Properties

Label 2-552-8.5-c1-0-0
Degree $2$
Conductor $552$
Sign $-0.196 - 0.980i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.624 − 1.26i)2-s − i·3-s + (−1.21 − 1.58i)4-s + 3.50i·5-s + (−1.26 − 0.624i)6-s − 4.50·7-s + (−2.77 + 0.556i)8-s − 9-s + (4.45 + 2.19i)10-s − 4.67i·11-s + (−1.58 + 1.21i)12-s + 3.59i·13-s + (−2.81 + 5.71i)14-s + 3.50·15-s + (−1.02 + 3.86i)16-s − 5.72·17-s + ⋯
L(s)  = 1  + (0.441 − 0.897i)2-s − 0.577i·3-s + (−0.609 − 0.792i)4-s + 1.56i·5-s + (−0.517 − 0.255i)6-s − 1.70·7-s + (−0.980 + 0.196i)8-s − 0.333·9-s + (1.40 + 0.693i)10-s − 1.41i·11-s + (−0.457 + 0.351i)12-s + 0.996i·13-s + (−0.752 + 1.52i)14-s + 0.906·15-s + (−0.256 + 0.966i)16-s − 1.38·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.196 - 0.980i$
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: \((2,\ 552,\ (\ :1/2),\ -0.196 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0832207 + 0.101581i\)
\(L(\frac12)\) \(\approx\) \(0.0832207 + 0.101581i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.624 + 1.26i)T \)
3 \( 1 + iT \)
23 \( 1 + T \)
good5 \( 1 - 3.50iT - 5T^{2} \)
7 \( 1 + 4.50T + 7T^{2} \)
11 \( 1 + 4.67iT - 11T^{2} \)
13 \( 1 - 3.59iT - 13T^{2} \)
17 \( 1 + 5.72T + 17T^{2} \)
19 \( 1 - 2.98iT - 19T^{2} \)
29 \( 1 - 7.72iT - 29T^{2} \)
31 \( 1 + 0.242T + 31T^{2} \)
37 \( 1 + 10.8iT - 37T^{2} \)
41 \( 1 - 1.48T + 41T^{2} \)
43 \( 1 + 4.67iT - 43T^{2} \)
47 \( 1 + 9.11T + 47T^{2} \)
53 \( 1 + 12.0iT - 53T^{2} \)
59 \( 1 - 2.22iT - 59T^{2} \)
61 \( 1 + 1.86iT - 61T^{2} \)
67 \( 1 - 2.80iT - 67T^{2} \)
71 \( 1 + 7.46T + 71T^{2} \)
73 \( 1 + 2.34T + 73T^{2} \)
79 \( 1 + 2.09T + 79T^{2} \)
83 \( 1 - 5.94iT - 83T^{2} \)
89 \( 1 + 10.9T + 89T^{2} \)
97 \( 1 - 17.6T + 97T^{2} \)
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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

−11.03447250049718927579583912058, −10.46327648953763458263936656829, −9.455989675848961353230971921650, −8.673809345518633193348057375932, −6.96889255512542871428821353854, −6.48997793279061835624744506075, −5.76539396998562437270860303997, −3.79320957577113921663433294000, −3.19501983359767221616205260443, −2.21340500443420944517445655549, 0.06059723818706547048028272233, 2.88514990342831128386586832914, 4.28565405013336676025799381536, 4.74911414345730086862957131381, 5.88563202554936224332263559725, 6.72082007316372958850634399762, 7.88754425082569245057747832901, 8.849663956173052077302605409043, 9.499544545953624793895125320775, 10.05379093314477796345403352570

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