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Properties

Label 2-177-3.2-c4-0-49
Degree $2$
Conductor $177$
Sign $0.985 + 0.171i$
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  + 3.56i·2-s + (−1.54 + 8.86i)3-s + 3.29·4-s − 32.9i·5-s + (−31.5 − 5.51i)6-s − 48.4·7-s + 68.7i·8-s + (−76.2 − 27.4i)9-s + 117.·10-s − 143. i·11-s + (−5.10 + 29.2i)12-s + 188.·13-s − 172. i·14-s + (292. + 51.0i)15-s − 192.·16-s − 314. i·17-s + ⋯
L(s)  = 1  + 0.890i·2-s + (−0.171 + 0.985i)3-s + 0.206·4-s − 1.31i·5-s + (−0.877 − 0.153i)6-s − 0.988·7-s + 1.07i·8-s + (−0.940 − 0.338i)9-s + 1.17·10-s − 1.18i·11-s + (−0.0354 + 0.203i)12-s + 1.11·13-s − 0.880i·14-s + (1.30 + 0.226i)15-s − 0.751·16-s − 1.08i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.171i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.985 + 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.985 + 0.171i$
Analytic conductor: \(18.2964\)
Root analytic conductor: \(4.27743\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :2),\ 0.985 + 0.171i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.365022598\)
\(L(\frac12)\) \(\approx\) \(1.365022598\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.54 - 8.86i)T \)
59 \( 1 + 453. iT \)
good2 \( 1 - 3.56iT - 16T^{2} \)
5 \( 1 + 32.9iT - 625T^{2} \)
7 \( 1 + 48.4T + 2.40e3T^{2} \)
11 \( 1 + 143. iT - 1.46e4T^{2} \)
13 \( 1 - 188.T + 2.85e4T^{2} \)
17 \( 1 + 314. iT - 8.35e4T^{2} \)
19 \( 1 + 1.38T + 1.30e5T^{2} \)
23 \( 1 + 193. iT - 2.79e5T^{2} \)
29 \( 1 + 1.26e3iT - 7.07e5T^{2} \)
31 \( 1 - 36.5T + 9.23e5T^{2} \)
37 \( 1 - 1.48e3T + 1.87e6T^{2} \)
41 \( 1 - 1.97e3iT - 2.82e6T^{2} \)
43 \( 1 - 203.T + 3.41e6T^{2} \)
47 \( 1 + 934. iT - 4.87e6T^{2} \)
53 \( 1 + 3.28e3iT - 7.89e6T^{2} \)
61 \( 1 + 5.62e3T + 1.38e7T^{2} \)
67 \( 1 + 4.79e3T + 2.01e7T^{2} \)
71 \( 1 - 3.64e3iT - 2.54e7T^{2} \)
73 \( 1 - 2.67e3T + 2.83e7T^{2} \)
79 \( 1 - 7.11e3T + 3.89e7T^{2} \)
83 \( 1 + 9.84e3iT - 4.74e7T^{2} \)
89 \( 1 + 3.14e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.30e4T + 8.85e7T^{2} \)
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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

−11.79665759539827133574138541402, −11.08436871745987038895019744365, −9.705445612497539126177301433904, −8.834111804042374344776795215417, −8.103139904265657339574217904755, −6.33092411481313351222565362966, −5.72295103354267104180008214488, −4.55054673065593722889005398743, −3.09190288862170523357187089957, −0.51733605244415433138135534636, 1.48694052883152801936584972256, 2.68549303017191198428840154726, 3.64493057348988687718129320829, 6.13714136136575658459841057852, 6.72084190734830272070403363472, 7.58183662111827895070721117559, 9.282634133807333152804552621488, 10.56524035270305828896380015767, 10.89347232737609619838776906905, 12.13967703563799932637547908077

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