Properties

Label 2-177-3.2-c2-0-16
Degree $2$
Conductor $177$
Sign $0.978 - 0.208i$
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  − 1.19i·2-s + (0.625 + 2.93i)3-s + 2.57·4-s + 0.192i·5-s + (3.49 − 0.745i)6-s + 6.00·7-s − 7.84i·8-s + (−8.21 + 3.66i)9-s + 0.229·10-s + 7.32i·11-s + (1.61 + 7.56i)12-s + 9.24·13-s − 7.15i·14-s + (−0.565 + 0.120i)15-s + 0.952·16-s + 9.92i·17-s + ⋯
L(s)  = 1  − 0.596i·2-s + (0.208 + 0.978i)3-s + 0.644·4-s + 0.0385i·5-s + (0.583 − 0.124i)6-s + 0.857·7-s − 0.980i·8-s + (−0.913 + 0.407i)9-s + 0.0229·10-s + 0.666i·11-s + (0.134 + 0.630i)12-s + 0.710·13-s − 0.511i·14-s + (−0.0376 + 0.00803i)15-s + 0.0595·16-s + 0.583i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.208i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.978 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.978 - 0.208i$
Analytic conductor: \(4.82290\)
Root analytic conductor: \(2.19611\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1),\ 0.978 - 0.208i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.98472 + 0.209073i\)
\(L(\frac12)\) \(\approx\) \(1.98472 + 0.209073i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.625 - 2.93i)T \)
59 \( 1 - 7.68iT \)
good2 \( 1 + 1.19iT - 4T^{2} \)
5 \( 1 - 0.192iT - 25T^{2} \)
7 \( 1 - 6.00T + 49T^{2} \)
11 \( 1 - 7.32iT - 121T^{2} \)
13 \( 1 - 9.24T + 169T^{2} \)
17 \( 1 - 9.92iT - 289T^{2} \)
19 \( 1 - 0.266T + 361T^{2} \)
23 \( 1 - 9.30iT - 529T^{2} \)
29 \( 1 + 28.6iT - 841T^{2} \)
31 \( 1 + 36.4T + 961T^{2} \)
37 \( 1 - 9.90T + 1.36e3T^{2} \)
41 \( 1 + 38.7iT - 1.68e3T^{2} \)
43 \( 1 + 39.2T + 1.84e3T^{2} \)
47 \( 1 - 17.9iT - 2.20e3T^{2} \)
53 \( 1 + 23.3iT - 2.80e3T^{2} \)
61 \( 1 + 71.8T + 3.72e3T^{2} \)
67 \( 1 + 58.1T + 4.48e3T^{2} \)
71 \( 1 + 25.1iT - 5.04e3T^{2} \)
73 \( 1 + 68.2T + 5.32e3T^{2} \)
79 \( 1 - 72.5T + 6.24e3T^{2} \)
83 \( 1 + 85.5iT - 6.88e3T^{2} \)
89 \( 1 + 94.8iT - 7.92e3T^{2} \)
97 \( 1 + 150.T + 9.40e3T^{2} \)
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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

−12.20877362293243902178463401030, −11.23404338739231408569380416778, −10.69580252220115429986535160179, −9.744684092766992133413201601412, −8.605714253826153048809458221918, −7.44175445716549536231010765044, −5.95367564699274296524839478186, −4.56296543038175641832257777421, −3.38610009889097709897150242225, −1.87293223188986844386301620149, 1.48027953298318755065302542307, 3.01295140450173346460469130173, 5.20577420431902506605405198682, 6.28900825343066237533960135685, 7.21212401940420339162825824550, 8.154050322175170946025002516314, 8.873281393295430553324467992263, 10.86717578174422997815893213564, 11.40014628945246386316765379566, 12.42734081639217668002478145486

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