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Properties

Label 1-179-179.20-r0-0-0
Degree $1$
Conductor $179$
Sign $-0.999 + 0.00430i$
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  + (0.227 − 0.973i)2-s + (0.938 − 0.345i)3-s + (−0.896 − 0.442i)4-s + (−0.969 − 0.244i)5-s + (−0.123 − 0.992i)6-s + (−0.999 + 0.0352i)7-s + (−0.635 + 0.772i)8-s + (0.760 − 0.648i)9-s + (−0.458 + 0.888i)10-s + (0.295 − 0.955i)11-s + (−0.994 − 0.105i)12-s + (−0.520 − 0.854i)13-s + (−0.192 + 0.981i)14-s + (−0.994 + 0.105i)15-s + (0.607 + 0.794i)16-s + (−0.925 − 0.378i)17-s + ⋯
L(s)  = 1  + (0.227 − 0.973i)2-s + (0.938 − 0.345i)3-s + (−0.896 − 0.442i)4-s + (−0.969 − 0.244i)5-s + (−0.123 − 0.992i)6-s + (−0.999 + 0.0352i)7-s + (−0.635 + 0.772i)8-s + (0.760 − 0.648i)9-s + (−0.458 + 0.888i)10-s + (0.295 − 0.955i)11-s + (−0.994 − 0.105i)12-s + (−0.520 − 0.854i)13-s + (−0.192 + 0.981i)14-s + (−0.994 + 0.105i)15-s + (0.607 + 0.794i)16-s + (−0.925 − 0.378i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $-0.999 + 0.00430i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ -0.999 + 0.00430i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002067423459 - 0.9596606951i\)
\(L(\frac12)\) \(\approx\) \(0.002067423459 - 0.9596606951i\)
\(L(1)\) \(\approx\) \(0.6482141873 - 0.7747441423i\)
\(L(1)\) \(\approx\) \(0.6482141873 - 0.7747441423i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (0.227 - 0.973i)T \)
3 \( 1 + (0.938 - 0.345i)T \)
5 \( 1 + (-0.969 - 0.244i)T \)
7 \( 1 + (-0.999 + 0.0352i)T \)
11 \( 1 + (0.295 - 0.955i)T \)
13 \( 1 + (-0.520 - 0.854i)T \)
17 \( 1 + (-0.925 - 0.378i)T \)
19 \( 1 + (-0.984 + 0.175i)T \)
23 \( 1 + (0.427 + 0.904i)T \)
29 \( 1 + (0.362 - 0.932i)T \)
31 \( 1 + (0.0881 - 0.996i)T \)
37 \( 1 + (0.362 + 0.932i)T \)
41 \( 1 + (0.844 - 0.535i)T \)
43 \( 1 + (0.880 - 0.474i)T \)
47 \( 1 + (-0.394 + 0.918i)T \)
53 \( 1 + (0.489 + 0.871i)T \)
59 \( 1 + (0.550 - 0.835i)T \)
61 \( 1 + (-0.688 - 0.725i)T \)
67 \( 1 + (-0.635 - 0.772i)T \)
71 \( 1 + (-0.825 + 0.564i)T \)
73 \( 1 + (0.960 - 0.278i)T \)
79 \( 1 + (-0.579 - 0.815i)T \)
83 \( 1 + (0.713 - 0.700i)T \)
89 \( 1 + (0.227 + 0.973i)T \)
97 \( 1 + (-0.329 - 0.944i)T \)
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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

−27.40366957912696832998072629355, −26.53722327406780821118918793642, −26.010909180718159790908076148815, −25.0906340538489093998838240244, −24.10040053348170125737750676247, −23.071741021417619196985608873341, −22.28469820663505722769596763898, −21.29414011961407769819566556015, −19.713221939486351302503471331860, −19.38478025505961511152865148195, −18.10564309981624075729040691297, −16.62467967042160921826269352209, −15.94605690113926659604136304831, −14.96502217363108873772458740713, −14.47387432527125174899354344439, −13.06047618107717739412871561762, −12.355354499824220617027938782585, −10.495454238022885084588509128391, −9.25024081586381759274943984301, −8.54992356023268629288848007026, −7.17292819568440276471137748310, −6.70147828324924841874198065695, −4.565837030146155574136795200279, −4.01096067732265672229626806135, −2.66903651568188858105398611207, 0.64230660054083535547663058103, 2.550385441132158698084597878037, 3.423582604554140006073487660803, 4.37328194401279800950607819218, 6.16566362271086937800093136925, 7.71058468507304328139343490320, 8.75165735079186926762533251614, 9.5799561501733421177966011371, 10.903111358006717151447606411313, 12.07107778206956977752584780704, 12.95319952049512319040594461084, 13.62778284022812109174414758074, 14.967034446501457013181751061068, 15.72056352770188819327397109084, 17.294259353698607108183079847682, 18.804165896218065015855672158202, 19.30509458270235485124818692352, 19.910777242798968987992570303207, 20.824492060037674350105569940517, 22.0129246823700652272212631170, 22.91833271405143810264082685523, 23.93598181280548625571653505830, 24.825766482476388066375126373079, 26.14419727788298787682934080362, 27.04709857649692337648197890368

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