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Properties

Label 2-177450-1.1-c1-0-64
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 + 3·11-s + 12-s − 14-s + 16-s + 18-s + 4·19-s − 21-s + 3·22-s − 9·23-s + 24-s + 27-s − 28-s − 6·29-s − 2·31-s + 32-s + 3·33-s + 36-s + 7·37-s + 4·38-s − 42-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 + 0.904·11-s + 0.288·12-s − 0.267·14-s + 1/4·16-s + 0.235·18-s + 0.917·19-s − 0.218·21-s + 0.639·22-s − 1.87·23-s + 0.204·24-s + 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.359·31-s + 0.176·32-s + 0.522·33-s + 1/6·36-s + 1.15·37-s + 0.648·38-s − 0.154·42-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

\(L(1)\) \(\approx\) \(4.519996429\)
\(L(\frac12)\) \(\approx\) \(4.519996429\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 11 T + p T^{2} \)
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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

−13.26170660440516, −12.71676640704985, −12.34550328991943, −11.77206732537127, −11.48775978294918, −10.93488701916686, −10.24797696545067, −9.832536312776256, −9.332749880436026, −9.096809246403547, −8.167709963276119, −7.889104624975364, −7.438787758038021, −6.704638415078997, −6.412065054331137, −5.827147156964198, −5.363251664618712, −4.602864275722995, −4.175137992238565, −3.583115656523818, −3.285288459834823, −2.606432230290378, −1.803159325084438, −1.556690746078171, −0.4886766628085699, 0.4886766628085699, 1.556690746078171, 1.803159325084438, 2.606432230290378, 3.285288459834823, 3.583115656523818, 4.175137992238565, 4.602864275722995, 5.363251664618712, 5.827147156964198, 6.412065054331137, 6.704638415078997, 7.438787758038021, 7.889104624975364, 8.167709963276119, 9.096809246403547, 9.332749880436026, 9.832536312776256, 10.24797696545067, 10.93488701916686, 11.48775978294918, 11.77206732537127, 12.34550328991943, 12.71676640704985, 13.26170660440516

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