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Properties

Label 2-552-8.5-c1-0-15
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + i·3-s + 2.00·4-s + 3.41i·5-s + 1.41i·6-s − 2·7-s + 2.82·8-s − 9-s + 4.82i·10-s + 4.24i·11-s + 2.00i·12-s − 6.82i·13-s − 2.82·14-s − 3.41·15-s + 4.00·16-s − 1.17·17-s + ⋯
L(s)  = 1  + 1.00·2-s + 0.577i·3-s + 1.00·4-s + 1.52i·5-s + 0.577i·6-s − 0.755·7-s + 1.00·8-s − 0.333·9-s + 1.52i·10-s + 1.27i·11-s + 0.577i·12-s − 1.89i·13-s − 0.755·14-s − 0.881·15-s + 1.00·16-s − 0.284·17-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

\(L(1)\) \(\approx\) \(1.75435 + 1.75435i\)
\(L(\frac12)\) \(\approx\) \(1.75435 + 1.75435i\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.41iT - 5T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 - 4.24iT - 11T^{2} \)
13 \( 1 + 6.82iT - 13T^{2} \)
17 \( 1 + 1.17T + 17T^{2} \)
19 \( 1 - 2.24iT - 19T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 - 10.8T + 31T^{2} \)
37 \( 1 + 3.07iT - 37T^{2} \)
41 \( 1 - 8.48T + 41T^{2} \)
43 \( 1 + 1.75iT - 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + 11.4iT - 53T^{2} \)
59 \( 1 - 5.65iT - 59T^{2} \)
61 \( 1 + 3.75iT - 61T^{2} \)
67 \( 1 - 9.07iT - 67T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 9.31T + 79T^{2} \)
83 \( 1 + 4.24iT - 83T^{2} \)
89 \( 1 - 5.17T + 89T^{2} \)
97 \( 1 - 3.65T + 97T^{2} \)
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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

−10.82191271414683686974322644206, −10.31608037311257845779600227586, −9.797623441190326816905810370918, −8.001313440855060543494464297698, −7.15584019859010278991969253448, −6.34415509488079099947188838304, −5.48398010754835501971789806793, −4.21831687025800726787452399863, −3.19184716384090121038526779994, −2.54381345219145338702351207672, 1.11396297898061111365393882851, 2.59241861904027987302320143141, 4.03816111179017422520714103861, 4.82291657399670731225564382914, 6.06432994051537544939110268122, 6.52865856739371681765162973505, 7.82158206505519119537488506044, 8.779318444734426049398374140783, 9.496288470788962529848890465651, 10.94674669365601985845462825066

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