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Properties

Label 2-552-69.68-c1-0-7
Degree $2$
Conductor $552$
Sign $0.998 + 0.0457i$
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  + (−0.356 − 1.69i)3-s − 0.441·5-s + 3.97i·7-s + (−2.74 + 1.20i)9-s + 4.64·11-s + 3.73·13-s + (0.157 + 0.748i)15-s + 0.441·17-s + 2.47i·19-s + (6.73 − 1.41i)21-s + (4.64 + 1.19i)23-s − 4.80·25-s + (3.02 + 4.22i)27-s − 6.41i·29-s + 4.71·31-s + ⋯
L(s)  = 1  + (−0.205 − 0.978i)3-s − 0.197·5-s + 1.50i·7-s + (−0.915 + 0.402i)9-s + 1.40·11-s + 1.03·13-s + (0.0406 + 0.193i)15-s + 0.107·17-s + 0.568i·19-s + (1.47 − 0.309i)21-s + (0.968 + 0.250i)23-s − 0.961·25-s + (0.582 + 0.813i)27-s − 1.19i·29-s + 0.846·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0457i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.998 + 0.0457i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.998 + 0.0457i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39531 - 0.0319284i\)
\(L(\frac12)\) \(\approx\) \(1.39531 - 0.0319284i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.356 + 1.69i)T \)
23 \( 1 + (-4.64 - 1.19i)T \)
good5 \( 1 + 0.441T + 5T^{2} \)
7 \( 1 - 3.97iT - 7T^{2} \)
11 \( 1 - 4.64T + 11T^{2} \)
13 \( 1 - 3.73T + 13T^{2} \)
17 \( 1 - 0.441T + 17T^{2} \)
19 \( 1 - 2.47iT - 19T^{2} \)
29 \( 1 + 6.41iT - 29T^{2} \)
31 \( 1 - 4.71T + 31T^{2} \)
37 \( 1 + 7.33iT - 37T^{2} \)
41 \( 1 - 0.365iT - 41T^{2} \)
43 \( 1 - 7.69iT - 43T^{2} \)
47 \( 1 + 2.36iT - 47T^{2} \)
53 \( 1 - 8.70T + 53T^{2} \)
59 \( 1 + 2.20iT - 59T^{2} \)
61 \( 1 - 13.1iT - 61T^{2} \)
67 \( 1 - 8.93iT - 67T^{2} \)
71 \( 1 - 8.22iT - 71T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 + 2.02iT - 79T^{2} \)
83 \( 1 + 11.2T + 83T^{2} \)
89 \( 1 + 17.3T + 89T^{2} \)
97 \( 1 + 5.82iT - 97T^{2} \)
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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

−11.26071706895491712834296903867, −9.718534299603164211370477139851, −8.782753268093293105337359279485, −8.267366381725844484188485458434, −7.08073242571430011035922372882, −6.08087237520988053994813380545, −5.65330342114740105200955674336, −3.99664388266501977874732526463, −2.62190522430456357811593211153, −1.36145812268064785181888150100, 1.02697618451288132871049900633, 3.40981765051781056354976467512, 4.00972465887367754181493624440, 4.93121974413667938206380617095, 6.32286039512481603584598457561, 7.01224119375113203464224882059, 8.317750153756769496466088184107, 9.140514359303430048983135386301, 9.996230179640186832945982576390, 10.85326596835701901909108272839

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