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Properties

Label 2-548-548.355-c0-0-0
Degree $2$
Conductor $548$
Sign $-0.0651 + 0.997i$
Analytic cond. $0.273487$
Root an. cond. $0.522960$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.602 − 0.798i)2-s + (−0.273 + 0.961i)4-s + (−0.840 − 0.634i)5-s + (0.932 − 0.361i)8-s + (0.982 + 0.183i)9-s + 1.05i·10-s + (1.29 − 1.42i)13-s + (−0.850 − 0.526i)16-s + (−1.73 − 0.673i)17-s + (−0.445 − 0.895i)18-s + (0.840 − 0.634i)20-s + (0.0296 + 0.104i)25-s + (−1.91 − 0.177i)26-s + (0.709 − 1.14i)29-s + (0.0922 + 0.995i)32-s + ⋯
L(s)  = 1  + (−0.602 − 0.798i)2-s + (−0.273 + 0.961i)4-s + (−0.840 − 0.634i)5-s + (0.932 − 0.361i)8-s + (0.982 + 0.183i)9-s + 1.05i·10-s + (1.29 − 1.42i)13-s + (−0.850 − 0.526i)16-s + (−1.73 − 0.673i)17-s + (−0.445 − 0.895i)18-s + (0.840 − 0.634i)20-s + (0.0296 + 0.104i)25-s + (−1.91 − 0.177i)26-s + (0.709 − 1.14i)29-s + (0.0922 + 0.995i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0651 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0651 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(548\)    =    \(2^{2} \cdot 137\)
Sign: $-0.0651 + 0.997i$
Analytic conductor: \(0.273487\)
Root analytic conductor: \(0.522960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{548} (355, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 548,\ (\ :0),\ -0.0651 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6021287075\)
\(L(\frac12)\) \(\approx\) \(0.6021287075\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.602 + 0.798i)T \)
137 \( 1 + (-0.739 + 0.673i)T \)
good3 \( 1 + (-0.982 - 0.183i)T^{2} \)
5 \( 1 + (0.840 + 0.634i)T + (0.273 + 0.961i)T^{2} \)
7 \( 1 + (-0.0922 - 0.995i)T^{2} \)
11 \( 1 + (0.850 + 0.526i)T^{2} \)
13 \( 1 + (-1.29 + 1.42i)T + (-0.0922 - 0.995i)T^{2} \)
17 \( 1 + (1.73 + 0.673i)T + (0.739 + 0.673i)T^{2} \)
19 \( 1 + (0.602 - 0.798i)T^{2} \)
23 \( 1 + (0.445 + 0.895i)T^{2} \)
29 \( 1 + (-0.709 + 1.14i)T + (-0.445 - 0.895i)T^{2} \)
31 \( 1 + (-0.602 - 0.798i)T^{2} \)
37 \( 1 - 1.20T + T^{2} \)
41 \( 1 - 1.79iT - T^{2} \)
43 \( 1 + (-0.602 + 0.798i)T^{2} \)
47 \( 1 + (0.932 + 0.361i)T^{2} \)
53 \( 1 + (-0.646 + 0.322i)T + (0.602 - 0.798i)T^{2} \)
59 \( 1 + (-0.932 - 0.361i)T^{2} \)
61 \( 1 + (1.67 - 0.312i)T + (0.932 - 0.361i)T^{2} \)
67 \( 1 + (0.0922 + 0.995i)T^{2} \)
71 \( 1 + (-0.850 + 0.526i)T^{2} \)
73 \( 1 + (1.45 - 1.32i)T + (0.0922 - 0.995i)T^{2} \)
79 \( 1 + (-0.982 + 0.183i)T^{2} \)
83 \( 1 + (0.739 - 0.673i)T^{2} \)
89 \( 1 + (-1.58 - 1.20i)T + (0.273 + 0.961i)T^{2} \)
97 \( 1 + (0.850 + 0.526i)T^{2} \)
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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

−10.85253317201909022434318900781, −9.978305964788891475396408256653, −9.012201590020422831029658664194, −8.189580771248802183991155120555, −7.64375039632627843868227506763, −6.37250891652443242602310556576, −4.65594160006990872685668188299, −4.07477176484574849070760763397, −2.73073713909456076610971733093, −1.01583251176756705325608325569, 1.74815219376076173129479445944, 3.86000749495380991849741168707, 4.55347968963359416409091641159, 6.22252272140235341423364167249, 6.80539217017838994255716217105, 7.49922324290504665444610394589, 8.699139827677047872396813258612, 9.126413191113485550122505725596, 10.47652274830194766760544355486, 10.94499760059415030748288581318

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