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Properties

Label 2-177-59.58-c4-0-34
Degree $2$
Conductor $177$
Sign $0.995 - 0.0942i$
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  + 7.81i·2-s + 5.19·3-s − 45.1·4-s + 21.3·5-s + 40.6i·6-s − 30.7·7-s − 227. i·8-s + 27·9-s + 166. i·10-s − 207. i·11-s − 234.·12-s − 206. i·13-s − 240. i·14-s + 110.·15-s + 1.05e3·16-s − 391.·17-s + ⋯
L(s)  = 1  + 1.95i·2-s + 0.577·3-s − 2.82·4-s + 0.852·5-s + 1.12i·6-s − 0.627·7-s − 3.55i·8-s + 0.333·9-s + 1.66i·10-s − 1.71i·11-s − 1.62·12-s − 1.21i·13-s − 1.22i·14-s + 0.492·15-s + 4.13·16-s − 1.35·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0942i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.995 - 0.0942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.995 - 0.0942i$
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: \((2,\ 177,\ (\ :2),\ 0.995 - 0.0942i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.109860004\)
\(L(\frac12)\) \(\approx\) \(1.109860004\)
\(L(3)\) 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 \( 1 + (3.46e3 - 327. i)T \)
good2 \( 1 - 7.81iT - 16T^{2} \)
5 \( 1 - 21.3T + 625T^{2} \)
7 \( 1 + 30.7T + 2.40e3T^{2} \)
11 \( 1 + 207. iT - 1.46e4T^{2} \)
13 \( 1 + 206. iT - 2.85e4T^{2} \)
17 \( 1 + 391.T + 8.35e4T^{2} \)
19 \( 1 + 321.T + 1.30e5T^{2} \)
23 \( 1 + 287. iT - 2.79e5T^{2} \)
29 \( 1 - 1.02e3T + 7.07e5T^{2} \)
31 \( 1 + 560. iT - 9.23e5T^{2} \)
37 \( 1 - 1.25e3iT - 1.87e6T^{2} \)
41 \( 1 + 1.52e3T + 2.82e6T^{2} \)
43 \( 1 - 2.15e3iT - 3.41e6T^{2} \)
47 \( 1 - 245. iT - 4.87e6T^{2} \)
53 \( 1 + 1.14e3T + 7.89e6T^{2} \)
61 \( 1 - 518. iT - 1.38e7T^{2} \)
67 \( 1 + 2.49e3iT - 2.01e7T^{2} \)
71 \( 1 - 9.31e3T + 2.54e7T^{2} \)
73 \( 1 + 8.00e3iT - 2.83e7T^{2} \)
79 \( 1 - 5.13e3T + 3.89e7T^{2} \)
83 \( 1 + 1.34e4iT - 4.74e7T^{2} \)
89 \( 1 - 3.14e3iT - 6.27e7T^{2} \)
97 \( 1 + 5.32e3iT - 8.85e7T^{2} \)
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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

−12.84385634505631471724409649663, −10.54136048661674832408338399388, −9.525638657648156727439768284341, −8.604019689744277299474921746278, −8.036188241220763555864720810008, −6.41677497575799181450289999712, −6.14425770340172358910647443572, −4.76153810735689083761831655047, −3.22905699916780354865536122876, −0.37375963864992062320504644804, 1.81786551632156286109827105068, 2.33476818771027573391606817298, 3.93029890145078721301510822413, 4.82559824854618155285620764722, 6.74871267629684733939794495116, 8.612622741198397256804141332025, 9.467305258614447593409102799116, 9.900646746388006524715874260998, 10.90515124632062156372725912313, 12.14095104589940610231984535134

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