Properties

Degree $2$
Conductor $338130$
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 + 4-s + 5-s − 8-s − 10-s + 4·11-s + 13-s + 16-s + 4·19-s + 20-s − 4·22-s + 8·23-s + 25-s − 26-s + 6·29-s + 8·31-s − 32-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 + 52-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 + 0.277·13-s + 1/4·16-s + 0.917·19-s + 0.223·20-s − 0.852·22-s + 1.66·23-s + 1/5·25-s − 0.196·26-s + 1.11·29-s + 1.43·31-s − 0.176·32-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 + 0.138·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338130\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13 \cdot 17^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{338130} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 338130,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.967997593\)
\(L(\frac12)\) \(\approx\) \(3.967997593\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
17 \( 1 \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
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

−12.33430223569649, −12.14081415803122, −11.55305128466023, −11.22514418197734, −10.76650236411782, −10.20331670468529, −9.727704871584752, −9.457455803277867, −8.925280395324920, −8.599434362747268, −8.040947863910194, −7.538952051578344, −6.990909853326554, −6.473795244174459, −6.320741277645235, −5.669313636393833, −4.941810610679109, −4.687502245739592, −3.905372811531476, −3.323202029749588, −2.830768540901701, −2.327625444756518, −1.512874435366828, −0.9599893629507264, −0.7561483430953913, 0.7561483430953913, 0.9599893629507264, 1.512874435366828, 2.327625444756518, 2.830768540901701, 3.323202029749588, 3.905372811531476, 4.687502245739592, 4.941810610679109, 5.669313636393833, 6.320741277645235, 6.473795244174459, 6.990909853326554, 7.538952051578344, 8.040947863910194, 8.599434362747268, 8.925280395324920, 9.457455803277867, 9.727704871584752, 10.20331670468529, 10.76650236411782, 11.22514418197734, 11.55305128466023, 12.14081415803122, 12.33430223569649

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