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Properties

Label 2-177-59.58-c4-0-13
Degree $2$
Conductor $177$
Sign $-0.985 + 0.169i$
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  + 6.76i·2-s + 5.19·3-s − 29.7·4-s + 6.77·5-s + 35.1i·6-s + 45.8·7-s − 93.3i·8-s + 27·9-s + 45.8i·10-s + 152. i·11-s − 154.·12-s + 138. i·13-s + 310. i·14-s + 35.2·15-s + 155.·16-s + 61.1·17-s + ⋯
L(s)  = 1  + 1.69i·2-s + 0.577·3-s − 1.86·4-s + 0.271·5-s + 0.976i·6-s + 0.936·7-s − 1.45i·8-s + 0.333·9-s + 0.458i·10-s + 1.25i·11-s − 1.07·12-s + 0.819i·13-s + 1.58i·14-s + 0.156·15-s + 0.605·16-s + 0.211·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.985 + 0.169i$
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: \((2,\ 177,\ (\ :2),\ -0.985 + 0.169i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.113870443\)
\(L(\frac12)\) \(\approx\) \(2.113870443\)
\(L(3)\) 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 \( 1 + (-3.43e3 + 589. i)T \)
good2 \( 1 - 6.76iT - 16T^{2} \)
5 \( 1 - 6.77T + 625T^{2} \)
7 \( 1 - 45.8T + 2.40e3T^{2} \)
11 \( 1 - 152. iT - 1.46e4T^{2} \)
13 \( 1 - 138. iT - 2.85e4T^{2} \)
17 \( 1 - 61.1T + 8.35e4T^{2} \)
19 \( 1 + 184.T + 1.30e5T^{2} \)
23 \( 1 - 55.4iT - 2.79e5T^{2} \)
29 \( 1 - 290.T + 7.07e5T^{2} \)
31 \( 1 - 681. iT - 9.23e5T^{2} \)
37 \( 1 - 1.42e3iT - 1.87e6T^{2} \)
41 \( 1 - 1.53e3T + 2.82e6T^{2} \)
43 \( 1 + 404. iT - 3.41e6T^{2} \)
47 \( 1 + 1.60e3iT - 4.87e6T^{2} \)
53 \( 1 + 1.32e3T + 7.89e6T^{2} \)
61 \( 1 + 2.73e3iT - 1.38e7T^{2} \)
67 \( 1 - 2.22e3iT - 2.01e7T^{2} \)
71 \( 1 + 6.15e3T + 2.54e7T^{2} \)
73 \( 1 - 2.81e3iT - 2.83e7T^{2} \)
79 \( 1 - 7.74e3T + 3.89e7T^{2} \)
83 \( 1 + 648. iT - 4.74e7T^{2} \)
89 \( 1 + 5.75e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.51e3iT - 8.85e7T^{2} \)
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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

−12.96016894682730613805592301007, −11.68826961030466427571571135075, −10.08657331872503588592436784996, −9.123618322444117452187886686911, −8.203132569832091468439864044524, −7.38239953952220495823437993124, −6.46667844039019539211302174177, −5.08302523563265057711399221877, −4.24790525578958557366658861265, −1.90608686966051787898974973254, 0.75392202189330378490113789258, 2.07192023664359082155583246494, 3.19907226758921591891741495454, 4.35416747983684999060575143687, 5.76703815251723084124717386390, 7.86227397512569918393928338397, 8.688831459505618321862504937627, 9.697127348474084039615746000509, 10.72129527302385438199196030209, 11.30094880753467862725063203322

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