Properties

Label 2-177-59.17-c1-0-6
Degree $2$
Conductor $177$
Sign $0.503 + 0.864i$
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  + (−1.39 + 2.05i)2-s + (0.0541 + 0.998i)3-s + (−1.53 − 3.85i)4-s + (−1.61 − 0.748i)5-s + (−2.12 − 1.27i)6-s + (−0.975 − 3.51i)7-s + (5.20 + 1.14i)8-s + (−0.994 + 0.108i)9-s + (3.78 − 2.27i)10-s + (−2.09 − 1.98i)11-s + (3.76 − 1.74i)12-s + (−5.08 − 0.553i)13-s + (8.57 + 2.88i)14-s + (0.659 − 1.65i)15-s + (−3.57 + 3.38i)16-s + (−1.42 + 5.13i)17-s + ⋯
L(s)  = 1  + (−0.984 + 1.45i)2-s + (0.0312 + 0.576i)3-s + (−0.768 − 1.92i)4-s + (−0.723 − 0.334i)5-s + (−0.867 − 0.521i)6-s + (−0.368 − 1.32i)7-s + (1.84 + 0.405i)8-s + (−0.331 + 0.0360i)9-s + (1.19 − 0.720i)10-s + (−0.632 − 0.598i)11-s + (1.08 − 0.503i)12-s + (−1.41 − 0.153i)13-s + (2.29 + 0.771i)14-s + (0.170 − 0.427i)15-s + (−0.894 + 0.847i)16-s + (−0.345 + 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 + 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.503 + 0.864i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.503 + 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.140055 - 0.0805258i\)
\(L(\frac12)\) \(\approx\) \(0.140055 - 0.0805258i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0541 - 0.998i)T \)
59 \( 1 + (7.64 - 0.737i)T \)
good2 \( 1 + (1.39 - 2.05i)T + (-0.740 - 1.85i)T^{2} \)
5 \( 1 + (1.61 + 0.748i)T + (3.23 + 3.81i)T^{2} \)
7 \( 1 + (0.975 + 3.51i)T + (-5.99 + 3.60i)T^{2} \)
11 \( 1 + (2.09 + 1.98i)T + (0.595 + 10.9i)T^{2} \)
13 \( 1 + (5.08 + 0.553i)T + (12.6 + 2.79i)T^{2} \)
17 \( 1 + (1.42 - 5.13i)T + (-14.5 - 8.76i)T^{2} \)
19 \( 1 + (-3.59 - 2.73i)T + (5.08 + 18.3i)T^{2} \)
23 \( 1 + (0.197 + 0.372i)T + (-12.9 + 19.0i)T^{2} \)
29 \( 1 + (4.40 + 6.49i)T + (-10.7 + 26.9i)T^{2} \)
31 \( 1 + (2.97 - 2.25i)T + (8.29 - 29.8i)T^{2} \)
37 \( 1 + (-6.60 + 1.45i)T + (33.5 - 15.5i)T^{2} \)
41 \( 1 + (-3.63 + 6.85i)T + (-23.0 - 33.9i)T^{2} \)
43 \( 1 + (6.02 - 5.71i)T + (2.32 - 42.9i)T^{2} \)
47 \( 1 + (5.25 - 2.43i)T + (30.4 - 35.8i)T^{2} \)
53 \( 1 + (7.20 + 4.33i)T + (24.8 + 46.8i)T^{2} \)
61 \( 1 + (-4.56 + 6.72i)T + (-22.5 - 56.6i)T^{2} \)
67 \( 1 + (-3.44 - 0.757i)T + (60.8 + 28.1i)T^{2} \)
71 \( 1 + (-13.9 + 6.44i)T + (45.9 - 54.1i)T^{2} \)
73 \( 1 + (-9.42 - 3.17i)T + (58.1 + 44.1i)T^{2} \)
79 \( 1 + (0.101 - 1.86i)T + (-78.5 - 8.54i)T^{2} \)
83 \( 1 + (1.40 - 8.54i)T + (-78.6 - 26.5i)T^{2} \)
89 \( 1 + (2.39 + 3.52i)T + (-32.9 + 82.6i)T^{2} \)
97 \( 1 + (-12.8 + 4.32i)T + (77.2 - 58.7i)T^{2} \)
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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

−12.69274183546033483472726818634, −11.08674202930201845283678445811, −10.11126158165809051287234332848, −9.482660747219984694355853455304, −7.949337796741161739342648569086, −7.80490355771595389888392554496, −6.43847816458353635222477904901, −5.16588698081903473254167493051, −3.89271427789999512720747448184, −0.19196648862275071662776778704, 2.28330547577589761239007501019, 3.07465997128870438315658454570, 5.07581193767412647405674495028, 7.14847142760926188934756160620, 7.902288897259619763693845522726, 9.248515256015772257944672822592, 9.651803873790240310944193745214, 11.16442300541145045251961169713, 11.75694897426617619612708510671, 12.42797817197939339630780935673

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