Properties

Label 1-177-177.29-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: $\chi_{177} (29, \cdot )$
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

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

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