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Properties

Label 2-177-177.176-c3-0-37
Degree $2$
Conductor $177$
Sign $0.739 + 0.673i$
Analytic cond. $10.4433$
Root an. cond. $3.23161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.5 − 3.84i)3-s − 8·4-s + 7.68i·5-s + 29·7-s + (−2.49 − 26.8i)9-s + (−28 + 30.7i)12-s + (29.4 + 26.8i)15-s + 64·16-s − 61.4i·17-s + 119·19-s − 61.4i·20-s + (101.5 − 111. i)21-s + 66·25-s + (−112 − 84.4i)27-s − 232·28-s − 268. i·29-s + ⋯
L(s)  = 1  + (0.673 − 0.739i)3-s − 4-s + 0.687i·5-s + 1.56·7-s + (−0.0925 − 0.995i)9-s + (−0.673 + 0.739i)12-s + (0.507 + 0.462i)15-s + 16-s − 0.876i·17-s + 1.43·19-s − 0.687i·20-s + (1.05 − 1.15i)21-s + 0.528·25-s + (−0.798 − 0.602i)27-s − 1.56·28-s − 1.72i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.739 + 0.673i$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ 0.739 + 0.673i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.88458 - 0.729916i\)
\(L(\frac12)\) \(\approx\) \(1.88458 - 0.729916i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.5 + 3.84i)T \)
59 \( 1 - 453. iT \)
good2 \( 1 + 8T^{2} \)
5 \( 1 - 7.68iT - 125T^{2} \)
7 \( 1 - 29T + 343T^{2} \)
11 \( 1 + 1.33e3T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 + 61.4iT - 4.91e3T^{2} \)
19 \( 1 - 119T + 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 + 268. iT - 2.43e4T^{2} \)
31 \( 1 - 2.97e4T^{2} \)
37 \( 1 - 5.06e4T^{2} \)
41 \( 1 - 7.68iT - 6.89e4T^{2} \)
43 \( 1 - 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 - 698. iT - 1.48e5T^{2} \)
61 \( 1 - 2.26e5T^{2} \)
67 \( 1 - 3.00e5T^{2} \)
71 \( 1 - 1.18e3iT - 3.57e5T^{2} \)
73 \( 1 - 3.89e5T^{2} \)
79 \( 1 + 835T + 4.93e5T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 - 9.12e5T^{2} \)
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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

−12.09754900734853022406129674921, −11.32714976573786649359600167782, −9.928500073106134652354494539438, −8.918114637909309176958437661475, −7.952062087992253160956052167368, −7.27134637775394327836645570978, −5.59738740432383147152077682183, −4.31829873925310447425772956738, −2.79274501633716833900179369085, −1.09885709454615079615261044027, 1.44273421242412220127641180227, 3.57348280875314381547334040845, 4.82408733762506964925126528470, 5.22893986232767485877879661595, 7.70221821928741192311864170520, 8.480523625788813089939296677648, 9.098158396872551140033711569384, 10.20781784494847144436184958513, 11.22112249526144457312122795401, 12.49338607896575808882919387680

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