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Properties

Label 1-179-179.42-r0-0-0
Degree $1$
Conductor $179$
Sign $0.691 - 0.722i$
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  + (−0.896 − 0.442i)2-s + (0.760 − 0.648i)3-s + (0.607 + 0.794i)4-s + (0.880 + 0.474i)5-s + (−0.969 + 0.244i)6-s + (0.997 − 0.0705i)7-s + (−0.192 − 0.981i)8-s + (0.158 − 0.987i)9-s + (−0.579 − 0.815i)10-s + (−0.825 − 0.564i)11-s + (0.977 + 0.210i)12-s + (−0.458 + 0.888i)13-s + (−0.925 − 0.378i)14-s + (0.977 − 0.210i)15-s + (−0.261 + 0.965i)16-s + (0.713 + 0.700i)17-s + ⋯
L(s)  = 1  + (−0.896 − 0.442i)2-s + (0.760 − 0.648i)3-s + (0.607 + 0.794i)4-s + (0.880 + 0.474i)5-s + (−0.969 + 0.244i)6-s + (0.997 − 0.0705i)7-s + (−0.192 − 0.981i)8-s + (0.158 − 0.987i)9-s + (−0.579 − 0.815i)10-s + (−0.825 − 0.564i)11-s + (0.977 + 0.210i)12-s + (−0.458 + 0.888i)13-s + (−0.925 − 0.378i)14-s + (0.977 − 0.210i)15-s + (−0.261 + 0.965i)16-s + (0.713 + 0.700i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.691 - 0.722i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ 0.691 - 0.722i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.093744070 - 0.4669257297i\)
\(L(\frac12)\) \(\approx\) \(1.093744070 - 0.4669257297i\)
\(L(1)\) \(\approx\) \(1.030672374 - 0.3183153214i\)
\(L(1)\) \(\approx\) \(1.030672374 - 0.3183153214i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (-0.896 - 0.442i)T \)
3 \( 1 + (0.760 - 0.648i)T \)
5 \( 1 + (0.880 + 0.474i)T \)
7 \( 1 + (0.997 - 0.0705i)T \)
11 \( 1 + (-0.825 - 0.564i)T \)
13 \( 1 + (-0.458 + 0.888i)T \)
17 \( 1 + (0.713 + 0.700i)T \)
19 \( 1 + (0.938 - 0.345i)T \)
23 \( 1 + (-0.635 + 0.772i)T \)
29 \( 1 + (-0.737 - 0.675i)T \)
31 \( 1 + (-0.984 - 0.175i)T \)
37 \( 1 + (-0.737 + 0.675i)T \)
41 \( 1 + (0.427 - 0.904i)T \)
43 \( 1 + (0.550 - 0.835i)T \)
47 \( 1 + (-0.688 - 0.725i)T \)
53 \( 1 + (-0.520 + 0.854i)T \)
59 \( 1 + (-0.394 - 0.918i)T \)
61 \( 1 + (-0.0529 + 0.998i)T \)
67 \( 1 + (-0.192 + 0.981i)T \)
71 \( 1 + (0.362 - 0.932i)T \)
73 \( 1 + (0.844 - 0.535i)T \)
79 \( 1 + (-0.329 + 0.944i)T \)
83 \( 1 + (0.0176 - 0.999i)T \)
89 \( 1 + (-0.896 + 0.442i)T \)
97 \( 1 + (-0.783 + 0.621i)T \)
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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

−27.49040611389773281197462493123, −26.436755355311138384479187777042, −25.640458645653745418520750980810, −24.812352552898436154203578370935, −24.23111866915958493945290147050, −22.65148429519605213478239042104, −21.24308768794348399078334004835, −20.55893684849314191110222771364, −20.01879963799232493643809557881, −18.37239425761518979415692139261, −17.874659063970580623450500964030, −16.650412285530046487964128190404, −15.84724893259343905676593214135, −14.6521823866135866126571050295, −14.14570039684929746166240727241, −12.62416603510980030952957333779, −10.93879645979767896786582986772, −10.01704643485393047180862705489, −9.33361397016863527622207837501, −8.133891677314020884398896847293, −7.51159399760337526863928639253, −5.48924468538617207128075977637, −4.95031128225596540161487365782, −2.72425685139412943866075221034, −1.621906973360408031990774412384, 1.52690122797189877824312324692, 2.30795671892030473201487620979, 3.56928551275167595844767104389, 5.69798380734783545811430205395, 7.16291782855990982541940105517, 7.873447246530207332427068267698, 9.00180067460165964112479468601, 9.932315423204318146171240998271, 11.08350541589362928037882415947, 12.13113538316665492047749511823, 13.43109553088658882501664931034, 14.19291884993467077945219601111, 15.36758119689256449972783202961, 16.899845685710059106668968915598, 17.78916220874823494782078712912, 18.52559482948048819008783361593, 19.21751115281928650761272023209, 20.49541067015644889701100479492, 21.15702478186030369522198203444, 21.92958742118330444294667870356, 23.91034629919342478819983981053, 24.459266288561746703689474589658, 25.657058351006564406306010265347, 26.223266151011410621619795075805, 26.96787385959491827654977242713

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