Properties

Label 2-178-1.1-c1-0-4
Degree $2$
Conductor $178$
Sign $1$
Analytic cond. $1.42133$
Root an. cond. $1.19219$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 3·5-s + 6-s − 4·7-s + 8-s − 2·9-s + 3·10-s − 6·11-s + 12-s + 2·13-s − 4·14-s + 3·15-s + 16-s + 3·17-s − 2·18-s + 5·19-s + 3·20-s − 4·21-s − 6·22-s − 3·23-s + 24-s + 4·25-s + 2·26-s − 5·27-s − 4·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s − 2/3·9-s + 0.948·10-s − 1.80·11-s + 0.288·12-s + 0.554·13-s − 1.06·14-s + 0.774·15-s + 1/4·16-s + 0.727·17-s − 0.471·18-s + 1.14·19-s + 0.670·20-s − 0.872·21-s − 1.27·22-s − 0.625·23-s + 0.204·24-s + 4/5·25-s + 0.392·26-s − 0.962·27-s − 0.755·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(178\)    =    \(2 \cdot 89\)
Sign: $1$
Analytic conductor: \(1.42133\)
Root analytic conductor: \(1.19219\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 178,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.968033122\)
\(L(\frac12)\) \(\approx\) \(1.968033122\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
89 \( 1 + T \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−13.09971700535401348969711896103, −12.01521975837500572096911321539, −10.38183771626813550069949920480, −9.924983367076292701096052683568, −8.721731410942193674670586885226, −7.36410689745433254804873728837, −5.94744023022306454130283790935, −5.48197881161062206485371067956, −3.31534392384618004485979133214, −2.51994269176562419526281033451, 2.51994269176562419526281033451, 3.31534392384618004485979133214, 5.48197881161062206485371067956, 5.94744023022306454130283790935, 7.36410689745433254804873728837, 8.721731410942193674670586885226, 9.924983367076292701096052683568, 10.38183771626813550069949920480, 12.01521975837500572096911321539, 13.09971700535401348969711896103

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