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Properties

Label 1-179-179.16-r0-0-0
Degree $1$
Conductor $179$
Sign $0.701 + 0.712i$
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  + (0.990 + 0.140i)2-s + (−0.896 + 0.442i)3-s + (0.960 + 0.278i)4-s + (0.804 + 0.593i)5-s + (−0.949 + 0.312i)6-s + (0.550 − 0.835i)7-s + (0.911 + 0.411i)8-s + (0.607 − 0.794i)9-s + (0.713 + 0.700i)10-s + (−0.520 + 0.854i)11-s + (−0.984 + 0.175i)12-s + (−0.925 − 0.378i)13-s + (0.662 − 0.749i)14-s + (−0.984 − 0.175i)15-s + (0.844 + 0.535i)16-s + (−0.123 − 0.992i)17-s + ⋯
L(s)  = 1  + (0.990 + 0.140i)2-s + (−0.896 + 0.442i)3-s + (0.960 + 0.278i)4-s + (0.804 + 0.593i)5-s + (−0.949 + 0.312i)6-s + (0.550 − 0.835i)7-s + (0.911 + 0.411i)8-s + (0.607 − 0.794i)9-s + (0.713 + 0.700i)10-s + (−0.520 + 0.854i)11-s + (−0.984 + 0.175i)12-s + (−0.925 − 0.378i)13-s + (0.662 − 0.749i)14-s + (−0.984 − 0.175i)15-s + (0.844 + 0.535i)16-s + (−0.123 − 0.992i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.701 + 0.712i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ 0.701 + 0.712i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.694864353 + 0.7093865563i\)
\(L(\frac12)\) \(\approx\) \(1.694864353 + 0.7093865563i\)
\(L(1)\) \(\approx\) \(1.573333723 + 0.4291657770i\)
\(L(1)\) \(\approx\) \(1.573333723 + 0.4291657770i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (0.990 + 0.140i)T \)
3 \( 1 + (-0.896 + 0.442i)T \)
5 \( 1 + (0.804 + 0.593i)T \)
7 \( 1 + (0.550 - 0.835i)T \)
11 \( 1 + (-0.520 + 0.854i)T \)
13 \( 1 + (-0.925 - 0.378i)T \)
17 \( 1 + (-0.123 - 0.992i)T \)
19 \( 1 + (0.227 + 0.973i)T \)
23 \( 1 + (0.977 + 0.210i)T \)
29 \( 1 + (-0.579 - 0.815i)T \)
31 \( 1 + (-0.783 + 0.621i)T \)
37 \( 1 + (-0.579 + 0.815i)T \)
41 \( 1 + (-0.994 + 0.105i)T \)
43 \( 1 + (0.295 - 0.955i)T \)
47 \( 1 + (0.362 - 0.932i)T \)
53 \( 1 + (-0.192 - 0.981i)T \)
59 \( 1 + (-0.825 - 0.564i)T \)
61 \( 1 + (-0.737 - 0.675i)T \)
67 \( 1 + (0.911 - 0.411i)T \)
71 \( 1 + (-0.458 + 0.888i)T \)
73 \( 1 + (-0.0529 - 0.998i)T \)
79 \( 1 + (0.0176 + 0.999i)T \)
83 \( 1 + (-0.969 - 0.244i)T \)
89 \( 1 + (0.990 - 0.140i)T \)
97 \( 1 + (-0.999 - 0.0352i)T \)
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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

−27.6583783619602487182075642382, −25.97534258018609275781609327135, −24.68792913112876367287166193604, −24.319444901589224670981027912758, −23.61311318284546905653189075293, −22.06222231913529034102695868532, −21.789214882729823149860971607294, −20.92701846165021515048122665279, −19.509019742762614441823233578620, −18.47964391103257477480993321655, −17.28374937065057632554024850072, −16.52033234079020983921437143963, −15.36361799331347321656923603506, −14.18272213372688600271225250073, −13.05744546744341604367109184484, −12.516708348274936717423675198745, −11.392070066258751813866977992043, −10.589337540351388983055964650035, −9.01700854452855561303386733500, −7.46118672977619255807923215221, −6.15393273848436525589738095958, −5.396116551576978283440234194932, −4.679211680425446464027519195967, −2.585950980229891898330250249739, −1.50863464475910733540995038518, 1.86050466333138892156571653292, 3.411324249314626851477858519710, 4.855026198584992864807265949331, 5.38072259462371214728032603971, 6.82611462835785656670326074097, 7.44922611611774268800198541687, 9.847028770478580999175629099422, 10.50904551835520289239567671596, 11.501327016832599953035874266751, 12.58387512234432634412611893293, 13.66947736453027271297847974642, 14.682150669691955868195930539768, 15.46398151722152056651016822564, 16.84974710076834399590768313671, 17.34955862514295981548682254709, 18.469974705038700782172802202872, 20.34990720561550761128297526501, 20.892655780045735429017983059352, 21.908603889703486397425366075004, 22.75392176486023692866868698505, 23.26710996561455424167882401212, 24.43902838070213988473151831234, 25.369879517571254436221459348733, 26.54620571363573550441250337364, 27.343144374908126803843884255088

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