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Properties

Label 1-547-547.157-r0-0-0
Degree $1$
Conductor $547$
Sign $0.999 + 0.0117i$
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  + (0.641 + 0.767i)2-s + (−0.222 − 0.974i)3-s + (−0.177 + 0.984i)4-s + (0.863 − 0.504i)5-s + (0.605 − 0.795i)6-s + (−0.519 + 0.854i)7-s + (−0.868 + 0.495i)8-s + (−0.900 + 0.433i)9-s + (0.940 + 0.338i)10-s + (−0.0402 − 0.999i)11-s + (0.998 − 0.0460i)12-s + (0.365 − 0.930i)13-s + (−0.988 + 0.149i)14-s + (−0.684 − 0.729i)15-s + (−0.937 − 0.349i)16-s + (0.993 + 0.114i)17-s + ⋯
L(s)  = 1  + (0.641 + 0.767i)2-s + (−0.222 − 0.974i)3-s + (−0.177 + 0.984i)4-s + (0.863 − 0.504i)5-s + (0.605 − 0.795i)6-s + (−0.519 + 0.854i)7-s + (−0.868 + 0.495i)8-s + (−0.900 + 0.433i)9-s + (0.940 + 0.338i)10-s + (−0.0402 − 0.999i)11-s + (0.998 − 0.0460i)12-s + (0.365 − 0.930i)13-s + (−0.988 + 0.149i)14-s + (−0.684 − 0.729i)15-s + (−0.937 − 0.349i)16-s + (0.993 + 0.114i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0117i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.999 + 0.0117i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.999 + 0.0117i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.907411562 + 0.01123631973i\)
\(L(\frac12)\) \(\approx\) \(1.907411562 + 0.01123631973i\)
\(L(1)\) \(\approx\) \(1.459503070 + 0.1238445995i\)
\(L(1)\) \(\approx\) \(1.459503070 + 0.1238445995i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.641 + 0.767i)T \)
3 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (0.863 - 0.504i)T \)
7 \( 1 + (-0.519 + 0.854i)T \)
11 \( 1 + (-0.0402 - 0.999i)T \)
13 \( 1 + (0.365 - 0.930i)T \)
17 \( 1 + (0.993 + 0.114i)T \)
19 \( 1 + (0.995 + 0.0919i)T \)
23 \( 1 + (0.211 - 0.977i)T \)
29 \( 1 + (-0.0862 + 0.996i)T \)
31 \( 1 + (0.449 + 0.893i)T \)
37 \( 1 + (0.233 - 0.972i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.407 + 0.913i)T \)
47 \( 1 + (0.987 + 0.160i)T \)
53 \( 1 + (0.973 - 0.228i)T \)
59 \( 1 + (0.692 + 0.721i)T \)
61 \( 1 + (-0.244 - 0.969i)T \)
67 \( 1 + (-0.459 + 0.888i)T \)
71 \( 1 + (-0.667 - 0.744i)T \)
73 \( 1 + (-0.684 + 0.729i)T \)
79 \( 1 + (-0.994 - 0.103i)T \)
83 \( 1 + (0.428 - 0.903i)T \)
89 \( 1 + (0.962 - 0.272i)T \)
97 \( 1 + (-0.806 + 0.591i)T \)
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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

−23.0706427670306302441093718234, −22.44242085594165257239479921971, −21.75260598122098719373224471923, −20.80621650413042692293680301787, −20.520031726417607126218701198900, −19.39599677981425238774647863485, −18.46457429951307435271386888808, −17.4326585505794123842391499074, −16.66005685993599899747178322561, −15.53715531552303201462355963245, −14.771914255104602502593782230613, −13.83016431308820216698187247013, −13.4197500828092772727420721732, −11.99653326802455852899657072847, −11.3284410724330106870342425682, −10.19725915693512708333655681791, −9.87831459208457238807326183420, −9.23450583544415799667458905370, −7.28190434891157839324219578902, −6.24388897729290061167508184882, −5.41760849458111540128913878901, −4.38483501508902672636520063698, −3.57079551460841516597422993659, −2.62651372172384937420925522821, −1.27274670840414040889543539313, 0.98750275569217814847559929931, 2.5891995730405582652909172378, 3.29942279467495499643309123219, 5.21853919301984142125289828053, 5.67712060974114809561139909679, 6.27986172219977818161195657635, 7.403990489623747772408246207085, 8.48214689342550180173221477909, 8.95728488388261551344592431212, 10.464039449961971842864549935339, 11.83321818122032774551954818293, 12.5491580204601560623349256686, 13.09189048821123073904755114523, 13.94037812065517767928865116016, 14.617425290526899123261547300802, 16.07253467330575188259854767433, 16.41671213470932207216226041423, 17.49965888617677174930162015665, 18.18079954191653651242539335147, 18.871371221586263557904868636421, 20.163091026252222758052382103567, 21.109619785464305839580332093047, 21.960460095448489900961549272224, 22.61932399069754479463694437606, 23.46170317886483186811095966156

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