↑

Properties

Label 1-179-179.169-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

Related objects

Downloads

Learn more

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: $\chi_{179} (169, \cdot )$
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 \( 1 + (0.427 - 0.904i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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