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Properties

Label 1-179-179.149-r0-0-0
Degree $1$
Conductor $179$
Sign $-0.687 - 0.726i$
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  + (0.760 − 0.648i)2-s + (0.662 − 0.749i)3-s + (0.158 − 0.987i)4-s + (−0.999 + 0.0352i)5-s + (0.0176 − 0.999i)6-s + (0.227 − 0.973i)7-s + (−0.520 − 0.854i)8-s + (−0.123 − 0.992i)9-s + (−0.737 + 0.675i)10-s + (−0.394 + 0.918i)11-s + (−0.635 − 0.772i)12-s + (0.362 + 0.932i)13-s + (−0.458 − 0.888i)14-s + (−0.635 + 0.772i)15-s + (−0.949 − 0.312i)16-s + (−0.579 + 0.815i)17-s + ⋯
L(s)  = 1  + (0.760 − 0.648i)2-s + (0.662 − 0.749i)3-s + (0.158 − 0.987i)4-s + (−0.999 + 0.0352i)5-s + (0.0176 − 0.999i)6-s + (0.227 − 0.973i)7-s + (−0.520 − 0.854i)8-s + (−0.123 − 0.992i)9-s + (−0.737 + 0.675i)10-s + (−0.394 + 0.918i)11-s + (−0.635 − 0.772i)12-s + (0.362 + 0.932i)13-s + (−0.458 − 0.888i)14-s + (−0.635 + 0.772i)15-s + (−0.949 − 0.312i)16-s + (−0.579 + 0.815i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $-0.687 - 0.726i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ -0.687 - 0.726i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6844364405 - 1.590556976i\)
\(L(\frac12)\) \(\approx\) \(0.6844364405 - 1.590556976i\)
\(L(1)\) \(\approx\) \(1.136796002 - 1.085047746i\)
\(L(1)\) \(\approx\) \(1.136796002 - 1.085047746i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (0.760 - 0.648i)T \)
3 \( 1 + (0.662 - 0.749i)T \)
5 \( 1 + (-0.999 + 0.0352i)T \)
7 \( 1 + (0.227 - 0.973i)T \)
11 \( 1 + (-0.394 + 0.918i)T \)
13 \( 1 + (0.362 + 0.932i)T \)
17 \( 1 + (-0.579 + 0.815i)T \)
19 \( 1 + (0.911 - 0.411i)T \)
23 \( 1 + (0.489 - 0.871i)T \)
29 \( 1 + (-0.0529 - 0.998i)T \)
31 \( 1 + (0.977 + 0.210i)T \)
37 \( 1 + (-0.0529 + 0.998i)T \)
41 \( 1 + (-0.863 - 0.505i)T \)
43 \( 1 + (0.997 + 0.0705i)T \)
47 \( 1 + (0.960 - 0.278i)T \)
53 \( 1 + (-0.825 + 0.564i)T \)
59 \( 1 + (0.990 + 0.140i)T \)
61 \( 1 + (0.844 - 0.535i)T \)
67 \( 1 + (-0.520 + 0.854i)T \)
71 \( 1 + (-0.688 + 0.725i)T \)
73 \( 1 + (-0.261 + 0.965i)T \)
79 \( 1 + (0.0881 - 0.996i)T \)
83 \( 1 + (-0.329 + 0.944i)T \)
89 \( 1 + (0.760 + 0.648i)T \)
97 \( 1 + (-0.984 + 0.175i)T \)
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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

−27.15309966200368910754146855470, −26.92433525333770824054211404088, −25.595879330762042761368811294568, −24.84530083071084455954078918446, −24.00737954186235743514579270093, −22.74425250842822276263080435483, −22.10218907179534708677243372136, −21.03807536482106323367155852080, −20.31717868412461346680249694092, −19.0911731978908105544632546065, −17.91065046858298224127945260688, −16.25292303086830228840457102473, −15.78521152652179755838371890468, −15.1288526557650432083440370093, −14.08650814247346754259039868170, −13.05085367200562560493968846255, −11.780857153630738701724547011989, −10.93639063576464830561964722007, −9.07722681495569017669845515109, −8.28582869620527925691040324592, −7.46436653554089425729066693108, −5.660320847472496377771340337013, −4.86304296328285629657602258870, −3.47639416931501694859628743094, −2.83432340862683850602890919207, 1.12460235793634999651280195347, 2.515646461212275962210519400726, 3.84452551958582030831829400663, 4.58636128932731151120211331159, 6.59113344211367729662021043788, 7.337441947970620291048281133781, 8.617926622355263270709520131552, 10.04630326209378209584053664571, 11.25203592161734437067199946658, 12.11927699911314676556566207748, 13.12174994739722388030911738131, 13.942802033330434551672631432, 14.9160399670658515954399338802, 15.77029017342797584078158615239, 17.441362983778886231434883714274, 18.743358649518255419412608089995, 19.38179382711671327123041099090, 20.40871317487840665426487414274, 20.71078176092067312690068479821, 22.38331894328900742036435914150, 23.409475856950799306378980277385, 23.80263578669513253958053706641, 24.696826327810891426922527046185, 26.189398823158617655930866811229, 26.85090373062040497873429996486

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