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Properties

Label 1-179-179.13-r0-0-0
Degree $1$
Conductor $179$
Sign $-0.903 + 0.429i$
Analytic cond. $0.831272$
Root an. cond. $0.831272$
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  + (−0.635 − 0.772i)2-s + (0.489 + 0.871i)3-s + (−0.192 + 0.981i)4-s + (−0.737 + 0.675i)5-s + (0.362 − 0.932i)6-s + (−0.994 − 0.105i)7-s + (0.880 − 0.474i)8-s + (−0.520 + 0.854i)9-s + (0.990 + 0.140i)10-s + (−0.783 − 0.621i)11-s + (−0.949 + 0.312i)12-s + (0.997 + 0.0705i)13-s + (0.550 + 0.835i)14-s + (−0.949 − 0.312i)15-s + (−0.925 − 0.378i)16-s + (−0.394 + 0.918i)17-s + ⋯
L(s)  = 1  + (−0.635 − 0.772i)2-s + (0.489 + 0.871i)3-s + (−0.192 + 0.981i)4-s + (−0.737 + 0.675i)5-s + (0.362 − 0.932i)6-s + (−0.994 − 0.105i)7-s + (0.880 − 0.474i)8-s + (−0.520 + 0.854i)9-s + (0.990 + 0.140i)10-s + (−0.783 − 0.621i)11-s + (−0.949 + 0.312i)12-s + (0.997 + 0.0705i)13-s + (0.550 + 0.835i)14-s + (−0.949 − 0.312i)15-s + (−0.925 − 0.378i)16-s + (−0.394 + 0.918i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.903 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.903 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $-0.903 + 0.429i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ -0.903 + 0.429i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05902079271 + 0.2615501848i\)
\(L(\frac12)\) \(\approx\) \(0.05902079271 + 0.2615501848i\)
\(L(1)\) \(\approx\) \(0.5188679899 + 0.1140899624i\)
\(L(1)\) \(\approx\) \(0.5188679899 + 0.1140899624i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (-0.635 - 0.772i)T \)
3 \( 1 + (0.489 + 0.871i)T \)
5 \( 1 + (-0.737 + 0.675i)T \)
7 \( 1 + (-0.994 - 0.105i)T \)
11 \( 1 + (-0.783 - 0.621i)T \)
13 \( 1 + (0.997 + 0.0705i)T \)
17 \( 1 + (-0.394 + 0.918i)T \)
19 \( 1 + (-0.863 - 0.505i)T \)
23 \( 1 + (-0.969 + 0.244i)T \)
29 \( 1 + (-0.896 - 0.442i)T \)
31 \( 1 + (-0.261 - 0.965i)T \)
37 \( 1 + (-0.896 + 0.442i)T \)
41 \( 1 + (-0.123 + 0.992i)T \)
43 \( 1 + (0.0881 + 0.996i)T \)
47 \( 1 + (0.938 + 0.345i)T \)
53 \( 1 + (-0.999 + 0.0352i)T \)
59 \( 1 + (-0.984 + 0.175i)T \)
61 \( 1 + (0.760 + 0.648i)T \)
67 \( 1 + (0.880 + 0.474i)T \)
71 \( 1 + (0.227 - 0.973i)T \)
73 \( 1 + (0.662 + 0.749i)T \)
79 \( 1 + (0.960 + 0.278i)T \)
83 \( 1 + (-0.688 + 0.725i)T \)
89 \( 1 + (-0.635 + 0.772i)T \)
97 \( 1 + (0.844 - 0.535i)T \)
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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

−26.68617017934670679650129233361, −25.7413733223181200153681203643, −25.24840013798215694778884813855, −24.132812122128284429508265758900, −23.40335475431693100227435511602, −22.74730337146906873272675981987, −20.550121052861261607027409876517, −20.02418113370458535683771181382, −18.91189697595837105389822561950, −18.39896717304862426252444038628, −17.18526031774517114889270164539, −15.95472755595245891926626894031, −15.521221213802262663726742861682, −14.100650046653365147287285511027, −13.07442110517828439676714104721, −12.24335146803767767861244125005, −10.66193453123561141869375812452, −9.25236794586807719837637882051, −8.52660712059290095090450295022, −7.53053417450932450554540747762, −6.64644246621689403624968993501, −5.42127499620793101381772123481, −3.74826719872521226550723364715, −1.955193771603553861987692031580, −0.235334203719067941284925182481, 2.44223206954850892695417674624, 3.480536251629385674322396942185, 4.13040679654058061470210440403, 6.23637901202817277825248351018, 7.820452427337921568263759944266, 8.595320479072561023068918129240, 9.76621544160056002246278918099, 10.725142642991924280872338550882, 11.25729444224130301557355733699, 12.864942590954322165484406639479, 13.734667082531355718419178133502, 15.32932082981890349501504608366, 15.960439413272398491609654399083, 16.92552423015843374990526021487, 18.45103662020658310155706441906, 19.204719507558246061813497428481, 19.88662379799122449926988288668, 20.9152152781284082197354164576, 21.87396668677226655044834982353, 22.56710498080582878539504511785, 23.721668357579855795534178749270, 25.640356232757826515867521755896, 26.16750110267029987203021679124, 26.615124025765745774231412295882, 27.88757387809011456022169565659

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