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Properties

Label 1-179-179.56-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: $\chi_{179} (56, \cdot )$
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

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

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