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Properties

Label 2-177-59.58-c8-0-4
Degree $2$
Conductor $177$
Sign $0.972 - 0.234i$
Analytic cond. $72.1060$
Root an. cond. $8.49152$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 29.8i·2-s + 46.7·3-s − 637.·4-s − 247.·5-s − 1.39e3i·6-s + 4.05e3·7-s + 1.14e4i·8-s + 2.18e3·9-s + 7.40e3i·10-s − 2.11e4i·11-s − 2.98e4·12-s + 2.15e4i·13-s − 1.21e5i·14-s − 1.15e4·15-s + 1.77e5·16-s − 9.51e4·17-s + ⋯
L(s)  = 1  − 1.86i·2-s + 0.577·3-s − 2.49·4-s − 0.396·5-s − 1.07i·6-s + 1.68·7-s + 2.78i·8-s + 0.333·9-s + 0.740i·10-s − 1.44i·11-s − 1.43·12-s + 0.753i·13-s − 3.15i·14-s − 0.228·15-s + 2.71·16-s − 1.13·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.972 - 0.234i$
Analytic conductor: \(72.1060\)
Root analytic conductor: \(8.49152\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :4),\ 0.972 - 0.234i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.5713039689\)
\(L(\frac12)\) \(\approx\) \(0.5713039689\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 46.7T \)
59 \( 1 + (1.17e7 - 2.84e6i)T \)
good2 \( 1 + 29.8iT - 256T^{2} \)
5 \( 1 + 247.T + 3.90e5T^{2} \)
7 \( 1 - 4.05e3T + 5.76e6T^{2} \)
11 \( 1 + 2.11e4iT - 2.14e8T^{2} \)
13 \( 1 - 2.15e4iT - 8.15e8T^{2} \)
17 \( 1 + 9.51e4T + 6.97e9T^{2} \)
19 \( 1 + 4.16e4T + 1.69e10T^{2} \)
23 \( 1 - 4.92e5iT - 7.83e10T^{2} \)
29 \( 1 + 1.25e6T + 5.00e11T^{2} \)
31 \( 1 - 1.66e6iT - 8.52e11T^{2} \)
37 \( 1 + 8.89e5iT - 3.51e12T^{2} \)
41 \( 1 - 1.19e6T + 7.98e12T^{2} \)
43 \( 1 + 2.01e6iT - 1.16e13T^{2} \)
47 \( 1 + 2.76e6iT - 2.38e13T^{2} \)
53 \( 1 + 8.79e6T + 6.22e13T^{2} \)
61 \( 1 + 1.24e7iT - 1.91e14T^{2} \)
67 \( 1 - 2.43e7iT - 4.06e14T^{2} \)
71 \( 1 + 4.11e7T + 6.45e14T^{2} \)
73 \( 1 - 2.32e7iT - 8.06e14T^{2} \)
79 \( 1 - 2.79e6T + 1.51e15T^{2} \)
83 \( 1 + 2.03e6iT - 2.25e15T^{2} \)
89 \( 1 - 2.00e6iT - 3.93e15T^{2} \)
97 \( 1 + 3.90e7iT - 7.83e15T^{2} \)
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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

−11.33879356954799250386693791980, −10.67886188326349200635670837048, −9.177945392793291221193355093403, −8.665968744681360211496706102862, −7.67377394406575134527402248437, −5.35419221665538845866723176088, −4.22948032482097343920118531716, −3.44247401720549717552605226805, −2.03113306704803521536190786341, −1.37704538919963111978388087758, 0.12405813186327325799088564705, 1.98217397326696491889146437416, 4.36636513786579981857350338046, 4.60815453698563007601750048939, 6.04404928087002153944097872002, 7.40954562517012841676729223778, 7.81431039936789566263245984817, 8.635632194150678137485663411946, 9.651356281772271958162843490982, 11.05278107466329421330553920227

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