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Properties

Label 2-552-184.91-c1-0-28
Degree $2$
Conductor $552$
Sign $-0.817 + 0.575i$
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  + (−0.409 − 1.35i)2-s − 3-s + (−1.66 + 1.10i)4-s + 3.34·5-s + (0.409 + 1.35i)6-s − 4.25·7-s + (2.18 + 1.79i)8-s + 9-s + (−1.37 − 4.52i)10-s − 1.35i·11-s + (1.66 − 1.10i)12-s − 2.36i·13-s + (1.74 + 5.75i)14-s − 3.34·15-s + (1.53 − 3.69i)16-s − 1.28i·17-s + ⋯
L(s)  = 1  + (−0.289 − 0.957i)2-s − 0.577·3-s + (−0.832 + 0.554i)4-s + 1.49·5-s + (0.167 + 0.552i)6-s − 1.60·7-s + (0.772 + 0.635i)8-s + 0.333·9-s + (−0.433 − 1.43i)10-s − 0.409i·11-s + (0.480 − 0.320i)12-s − 0.656i·13-s + (0.465 + 1.53i)14-s − 0.863·15-s + (0.384 − 0.923i)16-s − 0.312i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.817 + 0.575i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.817 + 0.575i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.245028 - 0.774180i\)
\(L(\frac12)\) \(\approx\) \(0.245028 - 0.774180i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.409 + 1.35i)T \)
3 \( 1 + T \)
23 \( 1 + (-1.27 + 4.62i)T \)
good5 \( 1 - 3.34T + 5T^{2} \)
7 \( 1 + 4.25T + 7T^{2} \)
11 \( 1 + 1.35iT - 11T^{2} \)
13 \( 1 + 2.36iT - 13T^{2} \)
17 \( 1 + 1.28iT - 17T^{2} \)
19 \( 1 + 1.70iT - 19T^{2} \)
29 \( 1 + 7.24iT - 29T^{2} \)
31 \( 1 + 9.70iT - 31T^{2} \)
37 \( 1 + 3.06T + 37T^{2} \)
41 \( 1 + 2.62T + 41T^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 - 6.05iT - 47T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 - 7.81T + 59T^{2} \)
61 \( 1 + 6.82T + 61T^{2} \)
67 \( 1 + 7.98iT - 67T^{2} \)
71 \( 1 + 13.4iT - 71T^{2} \)
73 \( 1 - 6.03T + 73T^{2} \)
79 \( 1 - 14.0T + 79T^{2} \)
83 \( 1 + 0.0200iT - 83T^{2} \)
89 \( 1 - 2.91iT - 89T^{2} \)
97 \( 1 + 13.4iT - 97T^{2} \)
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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

−10.34986052819403622356520272920, −9.569936536749714500253185273021, −9.344149120415218851957245878340, −7.909014665142306154848044405522, −6.43935193588802246690156104537, −5.93573789274017363133085782771, −4.73072532541554090503710563561, −3.25212229612408792430356536837, −2.32344994392889834294232308064, −0.55943464611101410579727323034, 1.62334482583729929036402516985, 3.54031723576827538647458089540, 5.11942078014294099469101734517, 5.75903060535387211783507771029, 6.73828293534070700975407168861, 6.95861488559494101282694850834, 8.742317844542570973609771790579, 9.427818291549695526128840347424, 10.04471866760185187601827536097, 10.62440201563479588887501823771

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