Properties

Label 2-546-273.257-c1-0-32
Degree $2$
Conductor $546$
Sign $0.875 + 0.484i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
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  + 2-s + (1.73 + 0.0741i)3-s + 4-s + (−1.09 − 0.634i)5-s + (1.73 + 0.0741i)6-s + (0.151 − 2.64i)7-s + 8-s + (2.98 + 0.256i)9-s + (−1.09 − 0.634i)10-s + (2.57 − 4.46i)11-s + (1.73 + 0.0741i)12-s + (−2.55 + 2.53i)13-s + (0.151 − 2.64i)14-s + (−1.85 − 1.17i)15-s + 16-s − 5.00·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.999 + 0.0427i)3-s + 0.5·4-s + (−0.491 − 0.283i)5-s + (0.706 + 0.0302i)6-s + (0.0572 − 0.998i)7-s + 0.353·8-s + (0.996 + 0.0855i)9-s + (−0.347 − 0.200i)10-s + (0.776 − 1.34i)11-s + (0.499 + 0.0213i)12-s + (−0.709 + 0.704i)13-s + (0.0404 − 0.705i)14-s + (−0.478 − 0.304i)15-s + 0.250·16-s − 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.875 + 0.484i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.875 + 0.484i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.76554 - 0.713914i\)
\(L(\frac12)\) \(\approx\) \(2.76554 - 0.713914i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-1.73 - 0.0741i)T \)
7 \( 1 + (-0.151 + 2.64i)T \)
13 \( 1 + (2.55 - 2.53i)T \)
good5 \( 1 + (1.09 + 0.634i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.57 + 4.46i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 5.00T + 17T^{2} \)
19 \( 1 + (-3.30 - 5.71i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.70iT - 23T^{2} \)
29 \( 1 + (0.776 - 0.448i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.25 - 7.37i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.53iT - 37T^{2} \)
41 \( 1 + (-4.54 + 2.62i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.72 + 2.98i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.18 + 2.41i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.7 - 6.76i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 11.8iT - 59T^{2} \)
61 \( 1 + (-4.04 + 2.33i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.0 + 5.78i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.79 + 6.56i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.210 - 0.365i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.95 - 5.10i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.22iT - 83T^{2} \)
89 \( 1 - 8.30iT - 89T^{2} \)
97 \( 1 + (-6.21 + 10.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.80858122851078152727116083375, −9.844067970777488165998180716453, −8.846133305883490183857754184132, −8.004455268278923323562030644571, −7.16467333320631835489685272836, −6.25894798320124497581957064022, −4.67567792649286941141403624756, −3.96190105230076977285548376066, −3.13798166917281644138684295738, −1.47470250178415165700350607099, 2.13078198030104423971276757182, 2.89268113102455649582393751454, 4.20703751933381215831231667231, 4.95215052689185100857469923754, 6.46211446480808418498760918713, 7.26900393036792907586279452535, 8.043912984018314079576729370234, 9.268608173834954422409885537805, 9.699547056953578530290493552006, 11.13317793240622915188671948309

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