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Properties

Label 2-555-185.184-c1-0-8
Degree $2$
Conductor $555$
Sign $-0.874 - 0.484i$
Analytic cond. $4.43169$
Root an. cond. $2.10515$
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  − 1.81·2-s + i·3-s + 1.28·4-s + (−2.16 + 0.541i)5-s − 1.81i·6-s + 4.65i·7-s + 1.29·8-s − 9-s + (3.93 − 0.981i)10-s + 4.78·11-s + 1.28i·12-s + 5.66·13-s − 8.43i·14-s + (−0.541 − 2.16i)15-s − 4.92·16-s − 1.75·17-s + ⋯
L(s)  = 1  − 1.28·2-s + 0.577i·3-s + 0.641·4-s + (−0.970 + 0.242i)5-s − 0.739i·6-s + 1.75i·7-s + 0.459·8-s − 0.333·9-s + (1.24 − 0.310i)10-s + 1.44·11-s + 0.370i·12-s + 1.57·13-s − 2.25i·14-s + (−0.139 − 0.560i)15-s − 1.23·16-s − 0.426·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $-0.874 - 0.484i$
Analytic conductor: \(4.43169\)
Root analytic conductor: \(2.10515\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 555,\ (\ :1/2),\ -0.874 - 0.484i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.135046 + 0.522384i\)
\(L(\frac12)\) \(\approx\) \(0.135046 + 0.522384i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 + (2.16 - 0.541i)T \)
37 \( 1 + (4.14 - 4.44i)T \)
good2 \( 1 + 1.81T + 2T^{2} \)
7 \( 1 - 4.65iT - 7T^{2} \)
11 \( 1 - 4.78T + 11T^{2} \)
13 \( 1 - 5.66T + 13T^{2} \)
17 \( 1 + 1.75T + 17T^{2} \)
19 \( 1 - 4.16iT - 19T^{2} \)
23 \( 1 + 5.64T + 23T^{2} \)
29 \( 1 + 0.922iT - 29T^{2} \)
31 \( 1 - 6.40iT - 31T^{2} \)
41 \( 1 + 0.323T + 41T^{2} \)
43 \( 1 - 2.09T + 43T^{2} \)
47 \( 1 - 1.02iT - 47T^{2} \)
53 \( 1 + 10.1iT - 53T^{2} \)
59 \( 1 - 14.2iT - 59T^{2} \)
61 \( 1 + 7.21iT - 61T^{2} \)
67 \( 1 - 4.18iT - 67T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 + 13.7iT - 73T^{2} \)
79 \( 1 + 2.74iT - 79T^{2} \)
83 \( 1 - 9.59iT - 83T^{2} \)
89 \( 1 - 1.00iT - 89T^{2} \)
97 \( 1 - 3.50T + 97T^{2} \)
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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

−11.05468889486029739577249971221, −10.11857940102558320996445946664, −9.123009996221342647000645741844, −8.611815245264093470303731094964, −8.168705624571029462160516943436, −6.71663468018580420818136845887, −5.86579918727999856685163818019, −4.36955095652952172126030446661, −3.38413199755385802859061237286, −1.66782191167856649256553668152, 0.54967526377456922873411995761, 1.43380246486981796756437901163, 3.79997518113318132460129955767, 4.30793586277460886089528883146, 6.39135551929774951708459424824, 7.12302723581524238193415225282, 7.78883485026144495660935918462, 8.616198039557335962272814933429, 9.291120721960721112229337611102, 10.48321647912704104293149209920

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