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Properties

Label 2-555-185.184-c1-0-13
Degree $2$
Conductor $555$
Sign $0.998 + 0.0614i$
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  − 0.273·2-s + i·3-s − 1.92·4-s + (−1.93 + 1.12i)5-s − 0.273i·6-s − 3.71i·7-s + 1.07·8-s − 9-s + (0.528 − 0.308i)10-s − 2.69·11-s − 1.92i·12-s + 5.16·13-s + 1.01i·14-s + (−1.12 − 1.93i)15-s + 3.55·16-s + 5.80·17-s + ⋯
L(s)  = 1  − 0.193·2-s + 0.577i·3-s − 0.962·4-s + (−0.863 + 0.503i)5-s − 0.111i·6-s − 1.40i·7-s + 0.379·8-s − 0.333·9-s + (0.167 − 0.0975i)10-s − 0.813·11-s − 0.555i·12-s + 1.43·13-s + 0.271i·14-s + (−0.290 − 0.498i)15-s + 0.889·16-s + 1.40·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $0.998 + 0.0614i$
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: \((2,\ 555,\ (\ :1/2),\ 0.998 + 0.0614i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.860868 - 0.0264568i\)
\(L(\frac12)\) \(\approx\) \(0.860868 - 0.0264568i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.93 - 1.12i)T \)
37 \( 1 + (-3.38 + 5.05i)T \)
good2 \( 1 + 0.273T + 2T^{2} \)
7 \( 1 + 3.71iT - 7T^{2} \)
11 \( 1 + 2.69T + 11T^{2} \)
13 \( 1 - 5.16T + 13T^{2} \)
17 \( 1 - 5.80T + 17T^{2} \)
19 \( 1 - 2.25iT - 19T^{2} \)
23 \( 1 + 1.20T + 23T^{2} \)
29 \( 1 - 3.66iT - 29T^{2} \)
31 \( 1 + 0.112iT - 31T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 - 1.72T + 43T^{2} \)
47 \( 1 - 2.45iT - 47T^{2} \)
53 \( 1 + 10.3iT - 53T^{2} \)
59 \( 1 - 1.51iT - 59T^{2} \)
61 \( 1 + 12.0iT - 61T^{2} \)
67 \( 1 + 0.195iT - 67T^{2} \)
71 \( 1 - 7.28T + 71T^{2} \)
73 \( 1 + 6.82iT - 73T^{2} \)
79 \( 1 + 14.4iT - 79T^{2} \)
83 \( 1 - 0.679iT - 83T^{2} \)
89 \( 1 + 2.26iT - 89T^{2} \)
97 \( 1 - 16.6T + 97T^{2} \)
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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

−10.56578118596787783234679663915, −10.14371045407076526682383496304, −9.030137936906772092533443205209, −7.87685176172042813228288080762, −7.68411018157867454972903961770, −6.13221957492013616798360617959, −4.92702772153060371268777101091, −3.84996028411291031370977162760, −3.48445775201517244533690648258, −0.77429124686190973018542931618, 1.01382726633712268627156231762, 2.88312424893908875093768497974, 4.12714685540142716553400159052, 5.35077176474318452372023243039, 5.94528342904368870509385111821, 7.60782345333952760277425619490, 8.240054034620272291447231475541, 8.788998424013172090623469088596, 9.646909809716064057458264882362, 10.90075225793924592252253238817

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