Properties

Label 2-177-1.1-c5-0-43
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $28.3879$
Root an. cond. $5.32803$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.77·2-s − 9·3-s + 45.0·4-s − 17.5·5-s − 79.0·6-s + 12.9·7-s + 114.·8-s + 81·9-s − 153.·10-s − 434.·11-s − 405.·12-s − 637.·13-s + 113.·14-s + 157.·15-s − 433.·16-s − 850.·17-s + 711.·18-s + 1.82e3·19-s − 789.·20-s − 116.·21-s − 3.81e3·22-s + 715.·23-s − 1.03e3·24-s − 2.81e3·25-s − 5.59e3·26-s − 729·27-s + 584.·28-s + ⋯
L(s)  = 1  + 1.55·2-s − 0.577·3-s + 1.40·4-s − 0.313·5-s − 0.896·6-s + 0.0999·7-s + 0.634·8-s + 0.333·9-s − 0.486·10-s − 1.08·11-s − 0.813·12-s − 1.04·13-s + 0.155·14-s + 0.180·15-s − 0.423·16-s − 0.713·17-s + 0.517·18-s + 1.15·19-s − 0.441·20-s − 0.0576·21-s − 1.67·22-s + 0.282·23-s − 0.366·24-s − 0.901·25-s − 1.62·26-s − 0.192·27-s + 0.140·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(28.3879\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 9T \)
59 \( 1 + 3.48e3T \)
good2 \( 1 - 8.77T + 32T^{2} \)
5 \( 1 + 17.5T + 3.12e3T^{2} \)
7 \( 1 - 12.9T + 1.68e4T^{2} \)
11 \( 1 + 434.T + 1.61e5T^{2} \)
13 \( 1 + 637.T + 3.71e5T^{2} \)
17 \( 1 + 850.T + 1.41e6T^{2} \)
19 \( 1 - 1.82e3T + 2.47e6T^{2} \)
23 \( 1 - 715.T + 6.43e6T^{2} \)
29 \( 1 - 1.13e3T + 2.05e7T^{2} \)
31 \( 1 + 9.55e3T + 2.86e7T^{2} \)
37 \( 1 + 107.T + 6.93e7T^{2} \)
41 \( 1 + 6.61e3T + 1.15e8T^{2} \)
43 \( 1 + 2.99e3T + 1.47e8T^{2} \)
47 \( 1 - 1.01e4T + 2.29e8T^{2} \)
53 \( 1 - 1.10e4T + 4.18e8T^{2} \)
61 \( 1 - 2.29e3T + 8.44e8T^{2} \)
67 \( 1 - 2.35e4T + 1.35e9T^{2} \)
71 \( 1 - 5.60e4T + 1.80e9T^{2} \)
73 \( 1 - 5.65e4T + 2.07e9T^{2} \)
79 \( 1 - 6.57e4T + 3.07e9T^{2} \)
83 \( 1 + 1.36e4T + 3.93e9T^{2} \)
89 \( 1 - 7.76e4T + 5.58e9T^{2} \)
97 \( 1 - 4.10e4T + 8.58e9T^{2} \)
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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

−11.61493548979022311593746718068, −10.78034315170231749179858965665, −9.488692356861375857158177338623, −7.74942466121667149284571639176, −6.81199967031303685497503191057, −5.46076912963869279406233881866, −4.92952099801868788953954305570, −3.64197115376387266212205742914, −2.30085111527069423611777125621, 0, 2.30085111527069423611777125621, 3.64197115376387266212205742914, 4.92952099801868788953954305570, 5.46076912963869279406233881866, 6.81199967031303685497503191057, 7.74942466121667149284571639176, 9.488692356861375857158177338623, 10.78034315170231749179858965665, 11.61493548979022311593746718068

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