Properties

Label 2-177-3.2-c4-0-1
Degree $2$
Conductor $177$
Sign $-0.995 - 0.0987i$
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  + 3.43i·2-s + (0.889 − 8.95i)3-s + 4.21·4-s − 18.6i·5-s + (30.7 + 3.05i)6-s − 68.5·7-s + 69.3i·8-s + (−79.4 − 15.9i)9-s + 63.9·10-s − 69.0i·11-s + (3.74 − 37.7i)12-s − 25.1·13-s − 235. i·14-s + (−166. − 16.5i)15-s − 170.·16-s + 489. i·17-s + ⋯
L(s)  = 1  + 0.858i·2-s + (0.0987 − 0.995i)3-s + 0.263·4-s − 0.744i·5-s + (0.854 + 0.0848i)6-s − 1.39·7-s + 1.08i·8-s + (−0.980 − 0.196i)9-s + 0.639·10-s − 0.570i·11-s + (0.0260 − 0.261i)12-s − 0.148·13-s − 1.20i·14-s + (−0.741 − 0.0735i)15-s − 0.667·16-s + 1.69i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0987i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.995 - 0.0987i$
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: \((2,\ 177,\ (\ :2),\ -0.995 - 0.0987i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.1255764217\)
\(L(\frac12)\) \(\approx\) \(0.1255764217\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.889 + 8.95i)T \)
59 \( 1 + 453. iT \)
good2 \( 1 - 3.43iT - 16T^{2} \)
5 \( 1 + 18.6iT - 625T^{2} \)
7 \( 1 + 68.5T + 2.40e3T^{2} \)
11 \( 1 + 69.0iT - 1.46e4T^{2} \)
13 \( 1 + 25.1T + 2.85e4T^{2} \)
17 \( 1 - 489. iT - 8.35e4T^{2} \)
19 \( 1 + 352.T + 1.30e5T^{2} \)
23 \( 1 - 736. iT - 2.79e5T^{2} \)
29 \( 1 + 166. iT - 7.07e5T^{2} \)
31 \( 1 + 776.T + 9.23e5T^{2} \)
37 \( 1 + 432.T + 1.87e6T^{2} \)
41 \( 1 + 3.19e3iT - 2.82e6T^{2} \)
43 \( 1 + 2.34e3T + 3.41e6T^{2} \)
47 \( 1 - 1.23e3iT - 4.87e6T^{2} \)
53 \( 1 + 4.42e3iT - 7.89e6T^{2} \)
61 \( 1 + 499.T + 1.38e7T^{2} \)
67 \( 1 + 1.40e3T + 2.01e7T^{2} \)
71 \( 1 - 578. iT - 2.54e7T^{2} \)
73 \( 1 + 9.31e3T + 2.83e7T^{2} \)
79 \( 1 + 3.81e3T + 3.89e7T^{2} \)
83 \( 1 + 4.62e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.93e3iT - 6.27e7T^{2} \)
97 \( 1 - 5.69e3T + 8.85e7T^{2} \)
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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

−12.79333973176448238506361120337, −11.76846267418433656417213845324, −10.54265893175308609379321046308, −8.983753199955976584246305914957, −8.298117513450853965094656478149, −7.15049266703071012284523187202, −6.27580064396934496504521801062, −5.57246413313920720184021496087, −3.43677680321300900689855759656, −1.81648974795155288307015657447, 0.04105365442453899755933524853, 2.60408490333450332038534657994, 3.19785706720768338397441526437, 4.54172113589759831825157677999, 6.31843254414593829314885153247, 7.11251145307497715327778732655, 9.003814914296151693245788824636, 9.909266614379147506881368051556, 10.41512927659478529306640307086, 11.33628427326135098123483626123

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