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Properties

Label 2-552-8.5-c1-0-41
Degree $2$
Conductor $552$
Sign $-0.880 - 0.474i$
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  + (0.231 − 1.39i)2-s − i·3-s + (−1.89 − 0.646i)4-s − 2.83i·5-s + (−1.39 − 0.231i)6-s + 0.890·7-s + (−1.34 + 2.49i)8-s − 9-s + (−3.95 − 0.657i)10-s + 0.936i·11-s + (−0.646 + 1.89i)12-s − 5.48i·13-s + (0.206 − 1.24i)14-s − 2.83·15-s + (3.16 + 2.44i)16-s − 5.50·17-s + ⋯
L(s)  = 1  + (0.163 − 0.986i)2-s − 0.577i·3-s + (−0.946 − 0.323i)4-s − 1.26i·5-s + (−0.569 − 0.0946i)6-s + 0.336·7-s + (−0.474 + 0.880i)8-s − 0.333·9-s + (−1.25 − 0.207i)10-s + 0.282i·11-s + (−0.186 + 0.546i)12-s − 1.52i·13-s + (0.0551 − 0.331i)14-s − 0.732·15-s + (0.790 + 0.611i)16-s − 1.33·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.880 - 0.474i$
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: \((2,\ 552,\ (\ :1/2),\ -0.880 - 0.474i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.280903 + 1.11437i\)
\(L(\frac12)\) \(\approx\) \(0.280903 + 1.11437i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.231 + 1.39i)T \)
3 \( 1 + iT \)
23 \( 1 + T \)
good5 \( 1 + 2.83iT - 5T^{2} \)
7 \( 1 - 0.890T + 7T^{2} \)
11 \( 1 - 0.936iT - 11T^{2} \)
13 \( 1 + 5.48iT - 13T^{2} \)
17 \( 1 + 5.50T + 17T^{2} \)
19 \( 1 + 1.52iT - 19T^{2} \)
29 \( 1 - 9.39iT - 29T^{2} \)
31 \( 1 - 9.44T + 31T^{2} \)
37 \( 1 - 4.28iT - 37T^{2} \)
41 \( 1 - 0.0216T + 41T^{2} \)
43 \( 1 + 5.79iT - 43T^{2} \)
47 \( 1 + 3.89T + 47T^{2} \)
53 \( 1 + 2.58iT - 53T^{2} \)
59 \( 1 + 12.0iT - 59T^{2} \)
61 \( 1 + 12.2iT - 61T^{2} \)
67 \( 1 + 2.40iT - 67T^{2} \)
71 \( 1 - 1.87T + 71T^{2} \)
73 \( 1 + 4.05T + 73T^{2} \)
79 \( 1 - 15.1T + 79T^{2} \)
83 \( 1 + 3.92iT - 83T^{2} \)
89 \( 1 - 13.6T + 89T^{2} \)
97 \( 1 + 5.80T + 97T^{2} \)
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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

−10.42624744434742726700568164898, −9.392077501941196803661347553278, −8.516449081639781264905953400954, −8.043308201153206358107305896996, −6.49492041691156473883521772552, −5.14473729630175258956215978132, −4.74708052589911939446921324706, −3.22383374683775288567832322349, −1.87586009150655209258632350714, −0.63942923141201765558047653104, 2.57646114052978977269844271054, 3.98061186231179735647626014711, 4.62910056243475193285473631997, 6.13584693698721743059192767020, 6.57154184689547493710328936722, 7.60688388987320284668693927328, 8.576722340254486630871642566882, 9.449035282769119381207753336274, 10.28589079390817264735672079711, 11.28901206078689261190970253056

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