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Properties

Label 2-15210-1.1-c1-0-23
Degree $2$
Conductor $15210$
Sign $1$
Analytic cond. $121.452$
Root an. cond. $11.0205$
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 + 4-s + 5-s + 8-s + 10-s + 4·11-s + 16-s + 6·17-s − 4·19-s + 20-s + 4·22-s − 8·23-s + 25-s − 6·29-s + 8·31-s + 32-s + 6·34-s + 10·37-s − 4·38-s + 40-s − 6·41-s + 4·43-s + 4·44-s − 8·46-s − 7·49-s + 50-s + 10·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s + 1.20·11-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.223·20-s + 0.852·22-s − 1.66·23-s + 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.176·32-s + 1.02·34-s + 1.64·37-s − 0.648·38-s + 0.158·40-s − 0.937·41-s + 0.609·43-s + 0.603·44-s − 1.17·46-s − 49-s + 0.141·50-s + 1.37·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15210\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(121.452\)
Root analytic conductor: \(11.0205\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 15210,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.537578847\)
\(L(\frac12)\) \(\approx\) \(4.537578847\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
13 \( 1 \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−16.02744589391183, −15.35707886284630, −14.74226498225327, −14.26896153568186, −14.04401151254811, −13.19527403626557, −12.77125428892316, −12.08910623149958, −11.68181987910453, −11.16982813608990, −10.20528736720515, −9.929112174183667, −9.327556218858882, −8.403804187550112, −7.982663537926223, −7.167221005008779, −6.460342107298707, −6.004279909068484, −5.503699038726014, −4.563303514730514, −3.994263898282488, −3.422822323024543, −2.448009274845654, −1.766432098684533, −0.8566253145796813, 0.8566253145796813, 1.766432098684533, 2.448009274845654, 3.422822323024543, 3.994263898282488, 4.563303514730514, 5.503699038726014, 6.004279909068484, 6.460342107298707, 7.167221005008779, 7.982663537926223, 8.403804187550112, 9.327556218858882, 9.929112174183667, 10.20528736720515, 11.16982813608990, 11.68181987910453, 12.08910623149958, 12.77125428892316, 13.19527403626557, 14.04401151254811, 14.26896153568186, 14.74226498225327, 15.35707886284630, 16.02744589391183

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