Properties

Label 2-177-59.19-c1-0-3
Degree $2$
Conductor $177$
Sign $0.289 - 0.957i$
Analytic cond. $1.41335$
Root an. cond. $1.18884$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.454 + 0.856i)2-s + (−0.725 + 0.687i)3-s + (0.595 + 0.877i)4-s + (3.60 − 0.792i)5-s + (−0.259 − 0.933i)6-s + (1.56 − 1.19i)7-s + (−2.94 + 0.320i)8-s + (0.0541 − 0.998i)9-s + (−0.956 + 3.44i)10-s + (−0.731 − 1.83i)11-s + (−1.03 − 0.227i)12-s + (0.326 + 6.01i)13-s + (0.309 + 1.88i)14-s + (−2.06 + 3.05i)15-s + (0.279 − 0.701i)16-s + (−1.98 − 1.50i)17-s + ⋯
L(s)  = 1  + (−0.321 + 0.605i)2-s + (−0.419 + 0.397i)3-s + (0.297 + 0.438i)4-s + (1.61 − 0.354i)5-s + (−0.105 − 0.381i)6-s + (0.592 − 0.450i)7-s + (−1.04 + 0.113i)8-s + (0.0180 − 0.332i)9-s + (−0.302 + 1.08i)10-s + (−0.220 − 0.553i)11-s + (−0.298 − 0.0658i)12-s + (0.0904 + 1.66i)13-s + (0.0825 + 0.503i)14-s + (−0.534 + 0.788i)15-s + (0.0698 − 0.175i)16-s + (−0.480 − 0.365i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.289 - 0.957i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.289 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.943639 + 0.700481i\)
\(L(\frac12)\) \(\approx\) \(0.943639 + 0.700481i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.725 - 0.687i)T \)
59 \( 1 + (-3.58 + 6.79i)T \)
good2 \( 1 + (0.454 - 0.856i)T + (-1.12 - 1.65i)T^{2} \)
5 \( 1 + (-3.60 + 0.792i)T + (4.53 - 2.09i)T^{2} \)
7 \( 1 + (-1.56 + 1.19i)T + (1.87 - 6.74i)T^{2} \)
11 \( 1 + (0.731 + 1.83i)T + (-7.98 + 7.56i)T^{2} \)
13 \( 1 + (-0.326 - 6.01i)T + (-12.9 + 1.40i)T^{2} \)
17 \( 1 + (1.98 + 1.50i)T + (4.54 + 16.3i)T^{2} \)
19 \( 1 + (0.510 - 0.172i)T + (15.1 - 11.4i)T^{2} \)
23 \( 1 + (2.30 - 1.38i)T + (10.7 - 20.3i)T^{2} \)
29 \( 1 + (4.68 + 8.84i)T + (-16.2 + 24.0i)T^{2} \)
31 \( 1 + (6.98 + 2.35i)T + (24.6 + 18.7i)T^{2} \)
37 \( 1 + (-8.07 - 0.878i)T + (36.1 + 7.95i)T^{2} \)
41 \( 1 + (-4.71 - 2.83i)T + (19.2 + 36.2i)T^{2} \)
43 \( 1 + (-1.43 + 3.59i)T + (-31.2 - 29.5i)T^{2} \)
47 \( 1 + (1.10 + 0.242i)T + (42.6 + 19.7i)T^{2} \)
53 \( 1 + (-2.40 - 8.65i)T + (-45.4 + 27.3i)T^{2} \)
61 \( 1 + (-4.54 + 8.56i)T + (-34.2 - 50.4i)T^{2} \)
67 \( 1 + (10.7 - 1.16i)T + (65.4 - 14.4i)T^{2} \)
71 \( 1 + (-1.03 - 0.227i)T + (64.4 + 29.8i)T^{2} \)
73 \( 1 + (1.59 + 9.75i)T + (-69.1 + 23.3i)T^{2} \)
79 \( 1 + (6.95 + 6.58i)T + (4.27 + 78.8i)T^{2} \)
83 \( 1 + (2.31 - 2.72i)T + (-13.4 - 81.9i)T^{2} \)
89 \( 1 + (-2.07 - 3.92i)T + (-49.9 + 73.6i)T^{2} \)
97 \( 1 + (1.55 - 9.46i)T + (-91.9 - 30.9i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.07232854223628689224292840148, −11.69478923148191316098059806827, −10.98619268791597282262669290898, −9.556468757448157698405658453326, −9.071979892609797635394445068229, −7.70186057853711594974160571470, −6.42850564819511128444432751165, −5.72689319239500275531296258552, −4.26890980700422301506410742969, −2.12691288700304963782268820533, 1.63440283136806531949831014629, 2.63876751714226950707468791032, 5.41213218390847033267286514583, 5.84964413685928000359191198716, 7.11812125372463736348193084229, 8.697200234377057493861024872037, 9.825817457399046311302635590369, 10.55345542373856618263787207918, 11.18811172789043879355005976703, 12.60118923780263370190756409242

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