Properties

Label 2-174-1.1-c1-0-0
Degree $2$
Conductor $174$
Sign $1$
Analytic cond. $1.38939$
Root an. cond. $1.17872$
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 − 3·7-s − 8-s + 9-s − 3·10-s + 6·11-s − 12-s + 3·14-s − 3·15-s + 16-s + 7·17-s − 18-s + 5·19-s + 3·20-s + 3·21-s − 6·22-s − 8·23-s + 24-s + 4·25-s − 27-s − 3·28-s + 29-s + 3·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 1.80·11-s − 0.288·12-s + 0.801·14-s − 0.774·15-s + 1/4·16-s + 1.69·17-s − 0.235·18-s + 1.14·19-s + 0.670·20-s + 0.654·21-s − 1.27·22-s − 1.66·23-s + 0.204·24-s + 4/5·25-s − 0.192·27-s − 0.566·28-s + 0.185·29-s + 0.547·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(174\)    =    \(2 \cdot 3 \cdot 29\)
Sign: $1$
Analytic conductor: \(1.38939\)
Root analytic conductor: \(1.17872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 174,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8687758076\)
\(L(\frac12)\) \(\approx\) \(0.8687758076\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
29 \( 1 - T \)
good5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−12.41495073006027608292886336535, −11.84547600245529311559567100217, −10.35936278010760181665887419466, −9.685450630995214363684292019821, −9.168917497360859090574738189737, −7.38150383670406160660461384354, −6.25882909168376670503818525224, −5.71658007760240133645615024953, −3.50678922223351589073161418512, −1.49567402999511499428987968793, 1.49567402999511499428987968793, 3.50678922223351589073161418512, 5.71658007760240133645615024953, 6.25882909168376670503818525224, 7.38150383670406160660461384354, 9.168917497360859090574738189737, 9.685450630995214363684292019821, 10.35936278010760181665887419466, 11.84547600245529311559567100217, 12.41495073006027608292886336535

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