Properties

Label 2-177-59.58-c4-0-21
Degree $2$
Conductor $177$
Sign $0.999 - 0.0296i$
Analytic cond. $18.2964$
Root an. cond. $4.27743$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.70i·2-s + 5.19·3-s − 29.0·4-s − 41.0·5-s + 34.8i·6-s − 6.70·7-s − 87.3i·8-s + 27·9-s − 275. i·10-s − 21.2i·11-s − 150.·12-s − 73.2i·13-s − 45.0i·14-s − 213.·15-s + 121.·16-s + 76.4·17-s + ⋯
L(s)  = 1  + 1.67i·2-s + 0.577·3-s − 1.81·4-s − 1.64·5-s + 0.968i·6-s − 0.136·7-s − 1.36i·8-s + 0.333·9-s − 2.75i·10-s − 0.175i·11-s − 1.04·12-s − 0.433i·13-s − 0.229i·14-s − 0.948·15-s + 0.476·16-s + 0.264·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0296i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.999 - 0.0296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.999 - 0.0296i$
Analytic conductor: \(18.2964\)
Root analytic conductor: \(4.27743\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :2),\ 0.999 - 0.0296i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.7054571617\)
\(L(\frac12)\) \(\approx\) \(0.7054571617\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 5.19T \)
59 \( 1 + (3.47e3 - 103. i)T \)
good2 \( 1 - 6.70iT - 16T^{2} \)
5 \( 1 + 41.0T + 625T^{2} \)
7 \( 1 + 6.70T + 2.40e3T^{2} \)
11 \( 1 + 21.2iT - 1.46e4T^{2} \)
13 \( 1 + 73.2iT - 2.85e4T^{2} \)
17 \( 1 - 76.4T + 8.35e4T^{2} \)
19 \( 1 - 439.T + 1.30e5T^{2} \)
23 \( 1 + 164. iT - 2.79e5T^{2} \)
29 \( 1 + 788.T + 7.07e5T^{2} \)
31 \( 1 + 754. iT - 9.23e5T^{2} \)
37 \( 1 + 1.70e3iT - 1.87e6T^{2} \)
41 \( 1 - 72.3T + 2.82e6T^{2} \)
43 \( 1 + 2.69e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.90e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.17e3T + 7.89e6T^{2} \)
61 \( 1 + 2.04e3iT - 1.38e7T^{2} \)
67 \( 1 + 796. iT - 2.01e7T^{2} \)
71 \( 1 + 4.86e3T + 2.54e7T^{2} \)
73 \( 1 + 1.82e3iT - 2.83e7T^{2} \)
79 \( 1 + 7.08e3T + 3.89e7T^{2} \)
83 \( 1 + 4.42e3iT - 4.74e7T^{2} \)
89 \( 1 + 8.04e3iT - 6.27e7T^{2} \)
97 \( 1 - 2.35e3iT - 8.85e7T^{2} \)
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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

−12.22875472034522738572088003112, −11.06990772110068775924446477697, −9.471189483372720571445096580718, −8.499749553312313997021103328668, −7.65494938576523318247347918845, −7.24509628476153407713961795950, −5.74611850972384889194009213973, −4.45214855674206408660852904845, −3.38146134537142875668586876463, −0.27624643458435111836920643575, 1.28486212299728859492038873172, 3.00925121966453451005116347655, 3.73540298211648157604697818162, 4.77662130215188206296358679645, 7.16794056578331659855671690960, 8.187056905157326816277693901706, 9.215963561308187624750459195632, 10.14510782661774794594002586086, 11.34727743111348930471036037789, 11.79032342084006228231928489410

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