Properties

Label 2-177-59.58-c4-0-28
Degree $2$
Conductor $177$
Sign $-0.0950 + 0.995i$
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  − 4.96i·2-s + 5.19·3-s − 8.66·4-s + 41.3·5-s − 25.8i·6-s + 1.08·7-s − 36.4i·8-s + 27·9-s − 205. i·10-s + 142. i·11-s − 45.0·12-s − 35.2i·13-s − 5.39i·14-s + 215.·15-s − 319.·16-s + 527.·17-s + ⋯
L(s)  = 1  − 1.24i·2-s + 0.577·3-s − 0.541·4-s + 1.65·5-s − 0.716i·6-s + 0.0221·7-s − 0.569i·8-s + 0.333·9-s − 2.05i·10-s + 1.17i·11-s − 0.312·12-s − 0.208i·13-s − 0.0275i·14-s + 0.956·15-s − 1.24·16-s + 1.82·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0950 + 0.995i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0950 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.0950 + 0.995i$
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: \((2,\ 177,\ (\ :2),\ -0.0950 + 0.995i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.267947296\)
\(L(\frac12)\) \(\approx\) \(3.267947296\)
\(L(3)\) 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 \( 1 + (-330. + 3.46e3i)T \)
good2 \( 1 + 4.96iT - 16T^{2} \)
5 \( 1 - 41.3T + 625T^{2} \)
7 \( 1 - 1.08T + 2.40e3T^{2} \)
11 \( 1 - 142. iT - 1.46e4T^{2} \)
13 \( 1 + 35.2iT - 2.85e4T^{2} \)
17 \( 1 - 527.T + 8.35e4T^{2} \)
19 \( 1 - 76.0T + 1.30e5T^{2} \)
23 \( 1 + 138. iT - 2.79e5T^{2} \)
29 \( 1 + 1.04e3T + 7.07e5T^{2} \)
31 \( 1 + 1.36e3iT - 9.23e5T^{2} \)
37 \( 1 - 2.07e3iT - 1.87e6T^{2} \)
41 \( 1 + 1.60e3T + 2.82e6T^{2} \)
43 \( 1 + 746. iT - 3.41e6T^{2} \)
47 \( 1 - 613. iT - 4.87e6T^{2} \)
53 \( 1 + 3.30e3T + 7.89e6T^{2} \)
61 \( 1 - 3.02e3iT - 1.38e7T^{2} \)
67 \( 1 + 3.67e3iT - 2.01e7T^{2} \)
71 \( 1 - 1.82e3T + 2.54e7T^{2} \)
73 \( 1 + 2.69e3iT - 2.83e7T^{2} \)
79 \( 1 - 3.08e3T + 3.89e7T^{2} \)
83 \( 1 + 7.95e3iT - 4.74e7T^{2} \)
89 \( 1 - 5.14e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.30e4iT - 8.85e7T^{2} \)
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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

−11.84138754925194782222609394936, −10.46385406560198691594202901980, −9.778452867781204628111282187666, −9.421303341871784465425480647943, −7.72253984636358720536870063078, −6.38355869158032543398491158679, −5.02260527739147452684303547296, −3.38404200986691277636208234198, −2.21517959781972745525100178322, −1.39179263910234966979364251037, 1.62652942034243090719973515921, 3.15551374908846648368915489884, 5.33327730399523438204165778643, 5.85950956016955291789136933881, 6.98723904542770035326670602885, 8.114848264233370778851425790980, 9.064759301950435933576996353048, 9.920727464169810026701389488517, 11.11401457841855406821443123368, 12.67033060898690236952145776730

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