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Properties

Label 1-547-547.111-r0-0-0
Degree $1$
Conductor $547$
Sign $0.471 + 0.881i$
Analytic cond. $2.54025$
Root an. cond. $2.54025$
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  + (−0.439 − 0.898i)2-s + (−0.222 − 0.974i)3-s + (−0.614 + 0.788i)4-s + (−0.311 + 0.950i)5-s + (−0.778 + 0.627i)6-s + (0.300 − 0.953i)7-s + (0.978 + 0.205i)8-s + (−0.900 + 0.433i)9-s + (0.990 − 0.137i)10-s + (−0.919 + 0.391i)11-s + (0.905 + 0.423i)12-s + (0.365 − 0.930i)13-s + (−0.988 + 0.149i)14-s + (0.995 + 0.0919i)15-s + (−0.244 − 0.969i)16-s + (−0.459 + 0.888i)17-s + ⋯
L(s)  = 1  + (−0.439 − 0.898i)2-s + (−0.222 − 0.974i)3-s + (−0.614 + 0.788i)4-s + (−0.311 + 0.950i)5-s + (−0.778 + 0.627i)6-s + (0.300 − 0.953i)7-s + (0.978 + 0.205i)8-s + (−0.900 + 0.433i)9-s + (0.990 − 0.137i)10-s + (−0.919 + 0.391i)11-s + (0.905 + 0.423i)12-s + (0.365 − 0.930i)13-s + (−0.988 + 0.149i)14-s + (0.995 + 0.0919i)15-s + (−0.244 − 0.969i)16-s + (−0.459 + 0.888i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.471 + 0.881i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.471 + 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1382907610 + 0.08283138658i\)
\(L(\frac12)\) \(\approx\) \(0.1382907610 + 0.08283138658i\)
\(L(1)\) \(\approx\) \(0.4672815241 - 0.2752095352i\)
\(L(1)\) \(\approx\) \(0.4672815241 - 0.2752095352i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.439 - 0.898i)T \)
3 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (-0.311 + 0.950i)T \)
7 \( 1 + (0.300 - 0.953i)T \)
11 \( 1 + (-0.919 + 0.391i)T \)
13 \( 1 + (0.365 - 0.930i)T \)
17 \( 1 + (-0.459 + 0.888i)T \)
19 \( 1 + (0.641 - 0.767i)T \)
23 \( 1 + (-0.944 - 0.327i)T \)
29 \( 1 + (-0.999 + 0.0345i)T \)
31 \( 1 + (-0.479 + 0.877i)T \)
37 \( 1 + (0.993 + 0.114i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.0632 + 0.997i)T \)
47 \( 1 + (-0.0402 + 0.999i)T \)
53 \( 1 + (-0.577 + 0.816i)T \)
59 \( 1 + (-0.200 + 0.979i)T \)
61 \( 1 + (0.00575 + 0.999i)T \)
67 \( 1 + (-0.819 + 0.572i)T \)
71 \( 1 + (-0.991 + 0.126i)T \)
73 \( 1 + (0.995 - 0.0919i)T \)
79 \( 1 + (-0.832 - 0.553i)T \)
83 \( 1 + (0.278 + 0.960i)T \)
89 \( 1 + (-0.868 + 0.495i)T \)
97 \( 1 + (-0.267 - 0.963i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.50137824662046129832893412436, −22.431881389523707864952143379, −21.641571134723594528920106344268, −20.71656767615078835734372351931, −20.03020696823894619760952100348, −18.67425627864955012534015283429, −18.21059648273751366219837894829, −17.02027696331402532715540770151, −16.257285192140574788579489055478, −15.88404809095865551485982508961, −15.11693617060224432967408160718, −14.12355458389469025334829141642, −13.172721442329396058325322609265, −11.8007464064138680729622866059, −11.18514751651520434185788538046, −9.79017266918113087840216708340, −9.25944609814123485875033154053, −8.44484076530379436072594211511, −7.71483364437460510842800918687, −6.10912631000575361831781323501, −5.4096573966124159543744229746, −4.75840101876919896056359215764, −3.7049684582147236108671749473, −1.91564898606378012247358098362, −0.10537320451151666476229067288, 1.308708557736549057289055073370, 2.44878802276901622917534561042, 3.306218837941777698312354913289, 4.49173457462319923681608451723, 5.8815943222437253237738299712, 7.236350113185361236270819592538, 7.64542220501588829359408602317, 8.48119619278234295651612164051, 10.02710701303866564195905223078, 10.81385828219488534985690764304, 11.18965298257128870389603786549, 12.36289021904380924901471634776, 13.161456065445890362423852243882, 13.78649731343329862296321223435, 14.83772213244230412810203170891, 16.112126525003724684018843662732, 17.2863458005274798596051471909, 18.04871017655345648793389178984, 18.23868097013511794540015971855, 19.44670013713034546474795301360, 19.97104163517790091604969014514, 20.70970223125317558924802412573, 22.03209528434230946733379819578, 22.61741974715746210571241542290, 23.453765255041755330358933333372

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