Properties

Label 2-177-59.58-c2-0-13
Degree $2$
Conductor $177$
Sign $0.0457 + 0.998i$
Analytic cond. $4.82290$
Root an. cond. $2.19611$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.86i·2-s − 1.73·3-s + 0.531·4-s − 9.04·5-s − 3.22i·6-s + 1.29·7-s + 8.43i·8-s + 2.99·9-s − 16.8i·10-s − 12.8i·11-s − 0.920·12-s − 23.5i·13-s + 2.42i·14-s + 15.6·15-s − 13.5·16-s − 10.1·17-s + ⋯
L(s)  = 1  + 0.931i·2-s − 0.577·3-s + 0.132·4-s − 1.80·5-s − 0.537i·6-s + 0.185·7-s + 1.05i·8-s + 0.333·9-s − 1.68i·10-s − 1.17i·11-s − 0.0767·12-s − 1.81i·13-s + 0.172i·14-s + 1.04·15-s − 0.849·16-s − 0.595·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0457 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0457 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.0457 + 0.998i$
Analytic conductor: \(4.82290\)
Root analytic conductor: \(2.19611\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1),\ 0.0457 + 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.209753 - 0.200372i\)
\(L(\frac12)\) \(\approx\) \(0.209753 - 0.200372i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73T \)
59 \( 1 + (2.69 + 58.9i)T \)
good2 \( 1 - 1.86iT - 4T^{2} \)
5 \( 1 + 9.04T + 25T^{2} \)
7 \( 1 - 1.29T + 49T^{2} \)
11 \( 1 + 12.8iT - 121T^{2} \)
13 \( 1 + 23.5iT - 169T^{2} \)
17 \( 1 + 10.1T + 289T^{2} \)
19 \( 1 + 23.3T + 361T^{2} \)
23 \( 1 + 9.25iT - 529T^{2} \)
29 \( 1 + 25.9T + 841T^{2} \)
31 \( 1 - 28.4iT - 961T^{2} \)
37 \( 1 + 22.3iT - 1.36e3T^{2} \)
41 \( 1 + 44.4T + 1.68e3T^{2} \)
43 \( 1 - 42.8iT - 1.84e3T^{2} \)
47 \( 1 - 30.6iT - 2.20e3T^{2} \)
53 \( 1 - 41.9T + 2.80e3T^{2} \)
61 \( 1 - 31.9iT - 3.72e3T^{2} \)
67 \( 1 + 94.0iT - 4.48e3T^{2} \)
71 \( 1 - 5.78T + 5.04e3T^{2} \)
73 \( 1 + 41.5iT - 5.32e3T^{2} \)
79 \( 1 + 105.T + 6.24e3T^{2} \)
83 \( 1 - 3.14iT - 6.88e3T^{2} \)
89 \( 1 - 21.8iT - 7.92e3T^{2} \)
97 \( 1 - 126. iT - 9.40e3T^{2} \)
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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

−12.13395404696092140747289702207, −10.99911299752020158513718278334, −10.86477612120840080561736129752, −8.479283276437994569572578231226, −8.067490833020680702931180406085, −7.03788624863901689051767298238, −5.94096087435491160762883725893, −4.75873199145830510021049819497, −3.27412775351234651140989889251, −0.17897300128400760021642409161, 1.94875150719745852764418749014, 3.93330706664305146857042348256, 4.47655420412993697077836011555, 6.74103325431471963586071523425, 7.30183920142475203881131681137, 8.765358573256974451951933740476, 10.04815705694856233542787014776, 11.19899505315678813277704865368, 11.60383607675572257253762023492, 12.22345620838470347758654911055

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