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Properties

Label 1-177-177.116-r1-0-0
Degree $1$
Conductor $177$
Sign $-0.366 + 0.930i$
Analytic cond. $19.0212$
Root an. cond. $19.0212$
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  + (0.994 + 0.108i)2-s + (0.976 + 0.214i)4-s + (−0.796 − 0.605i)5-s + (−0.370 + 0.928i)7-s + (0.947 + 0.319i)8-s + (−0.725 − 0.687i)10-s + (−0.907 + 0.419i)11-s + (−0.161 + 0.986i)13-s + (−0.468 + 0.883i)14-s + (0.907 + 0.419i)16-s + (0.370 + 0.928i)17-s + (−0.561 − 0.827i)19-s + (−0.647 − 0.762i)20-s + (−0.947 + 0.319i)22-s + (−0.0541 + 0.998i)23-s + ⋯
L(s)  = 1  + (0.994 + 0.108i)2-s + (0.976 + 0.214i)4-s + (−0.796 − 0.605i)5-s + (−0.370 + 0.928i)7-s + (0.947 + 0.319i)8-s + (−0.725 − 0.687i)10-s + (−0.907 + 0.419i)11-s + (−0.161 + 0.986i)13-s + (−0.468 + 0.883i)14-s + (0.907 + 0.419i)16-s + (0.370 + 0.928i)17-s + (−0.561 − 0.827i)19-s + (−0.647 − 0.762i)20-s + (−0.947 + 0.319i)22-s + (−0.0541 + 0.998i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.366 + 0.930i$
Analytic conductor: \(19.0212\)
Root analytic conductor: \(19.0212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (1:\ ),\ -0.366 + 0.930i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.166191972 + 1.713544859i\)
\(L(\frac12)\) \(\approx\) \(1.166191972 + 1.713544859i\)
\(L(1)\) \(\approx\) \(1.412614239 + 0.4676575029i\)
\(L(1)\) \(\approx\) \(1.412614239 + 0.4676575029i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (0.994 + 0.108i)T \)
5 \( 1 + (-0.796 - 0.605i)T \)
7 \( 1 + (-0.370 + 0.928i)T \)
11 \( 1 + (-0.907 + 0.419i)T \)
13 \( 1 + (-0.161 + 0.986i)T \)
17 \( 1 + (0.370 + 0.928i)T \)
19 \( 1 + (-0.561 - 0.827i)T \)
23 \( 1 + (-0.0541 + 0.998i)T \)
29 \( 1 + (0.994 - 0.108i)T \)
31 \( 1 + (-0.561 + 0.827i)T \)
37 \( 1 + (-0.947 + 0.319i)T \)
41 \( 1 + (-0.0541 - 0.998i)T \)
43 \( 1 + (0.907 + 0.419i)T \)
47 \( 1 + (-0.796 + 0.605i)T \)
53 \( 1 + (0.725 - 0.687i)T \)
61 \( 1 + (-0.994 - 0.108i)T \)
67 \( 1 + (-0.947 - 0.319i)T \)
71 \( 1 + (-0.796 + 0.605i)T \)
73 \( 1 + (0.468 - 0.883i)T \)
79 \( 1 + (0.647 + 0.762i)T \)
83 \( 1 + (0.856 - 0.515i)T \)
89 \( 1 + (0.994 - 0.108i)T \)
97 \( 1 + (0.468 + 0.883i)T \)
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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

−26.7771703324687867843829548128, −25.8707012061535498392023718318, −24.74256507747357310881971282067, −23.625613596943045715851630465489, −22.999109006860524110645323478913, −22.39790692682926668145418931540, −21.01679608477643998817853588884, −20.24201667555188665591166234796, −19.33386336577451728473963534746, −18.30027685436984112944054948607, −16.628822259656307563492790828289, −15.904768580656619103699496116186, −14.891505387489403462293448747146, −13.96665844815623917305385331503, −12.94930889041384132997953564610, −11.993190978579077872222847210, −10.69682824563744175539764815028, −10.30267005990737618875234197245, −8.00614328935584002390271486452, −7.24867050154697078451796264972, −6.07360466618396193244547275842, −4.71779376848323322901085711680, −3.554505023968119918225951175028, −2.69809872715529137522314558124, −0.50236783677429481720976806716, 1.90619359066878002982501055581, 3.26460654639593413965174916627, 4.518968739543326616372737279151, 5.42736507611881931626499271898, 6.72694344528209383232549194607, 7.89431556607043097111121851377, 9.0327671832554753303442174208, 10.64161533341849982313491022170, 11.86976637262214859242004375095, 12.46804607785966980695915993893, 13.382096088654659696640663982428, 14.78236906716943186614033131502, 15.61087328575873743290471193648, 16.2074758815499964015734342181, 17.482220158634507088133810569784, 19.14137763811527773089861302450, 19.67335352892116719855638494394, 21.07456511798947084161530264779, 21.54631161159703394730747278604, 22.78764787400328140268522907696, 23.72717783551495928819000698390, 24.1873910013594771856551428695, 25.48865389769782711167596580074, 26.11571374837175942652484096765, 27.70969623763092634246928570439

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