Properties

Degree $2$
Conductor $134310$
Sign $1$
Motivic weight $1$
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 + 5-s + 6-s − 8-s + 9-s − 10-s − 12-s + 2·13-s − 15-s + 16-s − 2·17-s − 18-s − 4·19-s + 20-s − 8·23-s + 24-s + 25-s − 2·26-s − 27-s + 2·29-s + 30-s + 8·31-s − 32-s + 2·34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.554·13-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.371·29-s + 0.182·30-s + 1.43·31-s − 0.176·32-s + 0.342·34-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 134310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(134310\)    =    \(2 \cdot 3 \cdot 5 \cdot 11^{2} \cdot 37\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{134310} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 134310,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9530880000\)
\(L(\frac12)\) \(\approx\) \(0.9530880000\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 \)
37 \( 1 - T \)
good7 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
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 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.43227070756756, −13.05707506752545, −12.22298639313451, −12.00990903730175, −11.47419342592801, −11.03983348969504, −10.36807641578284, −10.10138722809787, −9.816884333019369, −9.037755527583803, −8.519847265015090, −8.171356588636633, −7.735315651402977, −6.738873431598823, −6.481968396917895, −6.342587005603121, −5.451164976517314, −5.050334534998135, −4.338382605005790, −3.773272055921186, −3.117171036710264, −2.182872524902577, −1.950550546286944, −1.128446173673289, −0.3644896247031245, 0.3644896247031245, 1.128446173673289, 1.950550546286944, 2.182872524902577, 3.117171036710264, 3.773272055921186, 4.338382605005790, 5.050334534998135, 5.451164976517314, 6.342587005603121, 6.481968396917895, 6.738873431598823, 7.735315651402977, 8.171356588636633, 8.519847265015090, 9.037755527583803, 9.816884333019369, 10.10138722809787, 10.36807641578284, 11.03983348969504, 11.47419342592801, 12.00990903730175, 12.22298639313451, 13.05707506752545, 13.43227070756756

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