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Properties

Label 1-179-179.161-r0-0-0
Degree $1$
Conductor $179$
Sign $-0.0433 - 0.999i$
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  + (−0.192 − 0.981i)2-s + (−0.520 − 0.854i)3-s + (−0.925 + 0.378i)4-s + (0.0881 + 0.996i)5-s + (−0.737 + 0.675i)6-s + (0.977 − 0.210i)7-s + (0.550 + 0.835i)8-s + (−0.458 + 0.888i)9-s + (0.960 − 0.278i)10-s + (0.227 − 0.973i)11-s + (0.804 + 0.593i)12-s + (0.990 − 0.140i)13-s + (−0.394 − 0.918i)14-s + (0.804 − 0.593i)15-s + (0.713 − 0.700i)16-s + (−0.688 + 0.725i)17-s + ⋯
L(s)  = 1  + (−0.192 − 0.981i)2-s + (−0.520 − 0.854i)3-s + (−0.925 + 0.378i)4-s + (0.0881 + 0.996i)5-s + (−0.737 + 0.675i)6-s + (0.977 − 0.210i)7-s + (0.550 + 0.835i)8-s + (−0.458 + 0.888i)9-s + (0.960 − 0.278i)10-s + (0.227 − 0.973i)11-s + (0.804 + 0.593i)12-s + (0.990 − 0.140i)13-s + (−0.394 − 0.918i)14-s + (0.804 − 0.593i)15-s + (0.713 − 0.700i)16-s + (−0.688 + 0.725i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0433 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0433 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $-0.0433 - 0.999i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ -0.0433 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6300052709 - 0.6579220019i\)
\(L(\frac12)\) \(\approx\) \(0.6300052709 - 0.6579220019i\)
\(L(1)\) \(\approx\) \(0.7297749565 - 0.4834563363i\)
\(L(1)\) \(\approx\) \(0.7297749565 - 0.4834563363i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (-0.192 - 0.981i)T \)
3 \( 1 + (-0.520 - 0.854i)T \)
5 \( 1 + (0.0881 + 0.996i)T \)
7 \( 1 + (0.977 - 0.210i)T \)
11 \( 1 + (0.227 - 0.973i)T \)
13 \( 1 + (0.990 - 0.140i)T \)
17 \( 1 + (-0.688 + 0.725i)T \)
19 \( 1 + (0.489 - 0.871i)T \)
23 \( 1 + (0.880 + 0.474i)T \)
29 \( 1 + (0.607 - 0.794i)T \)
31 \( 1 + (-0.863 - 0.505i)T \)
37 \( 1 + (0.607 + 0.794i)T \)
41 \( 1 + (-0.969 + 0.244i)T \)
43 \( 1 + (-0.984 - 0.175i)T \)
47 \( 1 + (0.760 - 0.648i)T \)
53 \( 1 + (0.997 + 0.0705i)T \)
59 \( 1 + (0.938 + 0.345i)T \)
61 \( 1 + (0.158 - 0.987i)T \)
67 \( 1 + (0.550 - 0.835i)T \)
71 \( 1 + (-0.896 + 0.442i)T \)
73 \( 1 + (-0.123 - 0.992i)T \)
79 \( 1 + (0.844 - 0.535i)T \)
83 \( 1 + (-0.0529 + 0.998i)T \)
89 \( 1 + (-0.192 + 0.981i)T \)
97 \( 1 + (0.427 + 0.904i)T \)
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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

−27.403119784219998074835722666831, −26.93116285032924314747732518763, −25.488711116071863882779489421527, −24.86215223226160383066729638865, −23.70831318419776294950327571117, −23.07394309958150609354185352641, −21.95029429751084616745841857447, −20.86634282279408844801653372759, −20.15999142551952418582874592374, −18.2599786980678437627931801115, −17.69292682599535810792453974721, −16.67577588090626128001577319457, −15.9948535816935173822954211689, −15.04264265060150082500066896480, −14.108963712112277763905146026989, −12.74047477421454292580601422273, −11.56489163058691141069663621151, −10.2739031899426691522788564080, −9.09985162344931578785121458198, −8.53731751431429481870191188498, −7.041015715692552774292840682247, −5.62745049390026988965219014405, −4.892554083650411401880215485783, −4.06398672107699691907354077913, −1.29081869137746071932242658310, 1.10527688477471458581871529438, 2.33822716103337676327570602610, 3.65693858095250429335329976157, 5.24950480114350581980437696076, 6.528258257616888390172151447590, 7.82548582287553516884565920402, 8.74502548014700073045758142076, 10.5001229556701839602967780398, 11.23767278139346080893099404039, 11.64432551078224823026582345920, 13.42306396405927846526628170554, 13.63987163102472149925310979631, 15.04170101726672841101410279495, 16.868927584848043078379573728204, 17.73847534074609804738218139181, 18.40094554131812085591452427156, 19.16379771848037572623890189213, 20.17682986119714956020017252582, 21.53345383253358615670031673408, 22.09419890282115665147231463249, 23.258263969534116606659765774332, 23.90102446930917947067377782023, 25.24729172012638763481842629736, 26.43942111258814127497816773568, 27.21653591319736710545593307196

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