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Properties

Label 2-177-3.2-c4-0-23
Degree $2$
Conductor $177$
Sign $0.882 - 0.469i$
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  − 5.52i·2-s + (4.22 + 7.94i)3-s − 14.4·4-s + 14.8i·5-s + (43.8 − 23.3i)6-s + 83.7·7-s − 8.40i·8-s + (−45.2 + 67.1i)9-s + 82.1·10-s + 224. i·11-s + (−61.1 − 115. i)12-s − 309.·13-s − 462. i·14-s + (−118. + 62.8i)15-s − 278.·16-s + 375. i·17-s + ⋯
L(s)  = 1  − 1.38i·2-s + (0.469 + 0.882i)3-s − 0.904·4-s + 0.595i·5-s + (1.21 − 0.648i)6-s + 1.70·7-s − 0.131i·8-s + (−0.559 + 0.829i)9-s + 0.821·10-s + 1.85i·11-s + (−0.424 − 0.798i)12-s − 1.83·13-s − 2.35i·14-s + (−0.525 + 0.279i)15-s − 1.08·16-s + 1.29i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.469i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.882 - 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.882 - 0.469i$
Analytic conductor: \(18.2964\)
Root analytic conductor: \(4.27743\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :2),\ 0.882 - 0.469i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.149335253\)
\(L(\frac12)\) \(\approx\) \(2.149335253\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.22 - 7.94i)T \)
59 \( 1 + 453. iT \)
good2 \( 1 + 5.52iT - 16T^{2} \)
5 \( 1 - 14.8iT - 625T^{2} \)
7 \( 1 - 83.7T + 2.40e3T^{2} \)
11 \( 1 - 224. iT - 1.46e4T^{2} \)
13 \( 1 + 309.T + 2.85e4T^{2} \)
17 \( 1 - 375. iT - 8.35e4T^{2} \)
19 \( 1 + 139.T + 1.30e5T^{2} \)
23 \( 1 + 571. iT - 2.79e5T^{2} \)
29 \( 1 - 607. iT - 7.07e5T^{2} \)
31 \( 1 - 1.38e3T + 9.23e5T^{2} \)
37 \( 1 - 595.T + 1.87e6T^{2} \)
41 \( 1 + 742. iT - 2.82e6T^{2} \)
43 \( 1 - 292.T + 3.41e6T^{2} \)
47 \( 1 - 534. iT - 4.87e6T^{2} \)
53 \( 1 - 3.09e3iT - 7.89e6T^{2} \)
61 \( 1 + 42.3T + 1.38e7T^{2} \)
67 \( 1 - 1.85e3T + 2.01e7T^{2} \)
71 \( 1 + 3.82e3iT - 2.54e7T^{2} \)
73 \( 1 - 8.26e3T + 2.83e7T^{2} \)
79 \( 1 + 4.67e3T + 3.89e7T^{2} \)
83 \( 1 + 3.66e3iT - 4.74e7T^{2} \)
89 \( 1 + 2.16e3iT - 6.27e7T^{2} \)
97 \( 1 - 5.52e3T + 8.85e7T^{2} \)
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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

−12.00386541442096498631076006921, −10.78290651956255799960621694137, −10.40632993477185412613524960955, −9.535454336072660804385280529626, −8.255525050661716401513662383480, −7.13231022966295455904057188607, −4.73734619262574267087487827698, −4.40552254074190930704729216794, −2.59917014749775998411729127993, −1.90133853550352270699183362968, 0.74481136285264507728226068256, 2.49225278384356451542402249520, 4.80260437967547435250275323760, 5.56927567148987988936620184630, 6.91292234488082241270105279054, 7.930632761988102246613432212928, 8.273919252878059632012138838214, 9.305942268893341018265899155499, 11.36154828176583566067287896976, 11.86171171021859940090421811006

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