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Properties

Label 2-177450-1.1-c1-0-90
Degree $2$
Conductor $177450$
Sign $1$
Analytic cond. $1416.94$
Root an. cond. $37.6423$
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 − 6-s − 7-s + 8-s + 9-s + 4·11-s − 12-s − 14-s + 16-s − 2·17-s + 18-s − 4·19-s + 21-s + 4·22-s + 8·23-s − 24-s − 27-s − 28-s − 2·29-s + 32-s − 4·33-s − 2·34-s + 36-s + 6·37-s − 4·38-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.917·19-s + 0.218·21-s + 0.852·22-s + 1.66·23-s − 0.204·24-s − 0.192·27-s − 0.188·28-s − 0.371·29-s + 0.176·32-s − 0.696·33-s − 0.342·34-s + 1/6·36-s + 0.986·37-s − 0.648·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1416.94\)
Root analytic conductor: \(37.6423\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.072042515\)
\(L(\frac12)\) \(\approx\) \(4.072042515\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
13 \( 1 \)
good11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.10213581457161, −12.80164198961421, −12.17936125237680, −11.92027414769377, −11.25851454966361, −10.85235367280647, −10.71859238966980, −9.830002144603052, −9.329889628988577, −9.070321002316558, −8.379172630502863, −7.771791288703511, −7.052087956615225, −6.787279632784368, −6.400651150500246, −5.770823803621089, −5.434763270227510, −4.625209634561481, −4.292158281244598, −3.842121002605326, −3.165353863868842, −2.535195362129891, −1.932016639070809, −1.117720418944757, −0.5917376475862539, 0.5917376475862539, 1.117720418944757, 1.932016639070809, 2.535195362129891, 3.165353863868842, 3.842121002605326, 4.292158281244598, 4.625209634561481, 5.434763270227510, 5.770823803621089, 6.400651150500246, 6.787279632784368, 7.052087956615225, 7.771791288703511, 8.379172630502863, 9.070321002316558, 9.329889628988577, 9.830002144603052, 10.71859238966980, 10.85235367280647, 11.25851454966361, 11.92027414769377, 12.17936125237680, 12.80164198961421, 13.10213581457161

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