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Properties

Label 2-338130-1.1-c1-0-18
Degree $2$
Conductor $338130$
Sign $1$
Analytic cond. $2699.98$
Root an. cond. $51.9613$
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 − 4·7-s + 8-s − 10-s − 4·11-s − 13-s − 4·14-s + 16-s − 20-s − 4·22-s + 4·23-s + 25-s − 26-s − 4·28-s − 8·29-s + 5·31-s + 32-s + 4·35-s − 2·37-s − 40-s + 10·41-s + 10·43-s − 4·44-s + 4·46-s − 12·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s + 0.353·8-s − 0.316·10-s − 1.20·11-s − 0.277·13-s − 1.06·14-s + 1/4·16-s − 0.223·20-s − 0.852·22-s + 0.834·23-s + 1/5·25-s − 0.196·26-s − 0.755·28-s − 1.48·29-s + 0.898·31-s + 0.176·32-s + 0.676·35-s − 0.328·37-s − 0.158·40-s + 1.56·41-s + 1.52·43-s − 0.603·44-s + 0.589·46-s − 1.75·47-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: $1$
Analytic conductor: \(2699.98\)
Root analytic conductor: \(51.9613\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 338130,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.918909803\)
\(L(\frac12)\) \(\approx\) \(1.918909803\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 13 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
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

−12.65601910793929, −12.46890474056297, −11.65942783423738, −11.20434034486187, −10.92337072642965, −10.33955510854648, −9.887909014864096, −9.395156429823512, −9.134352525493640, −8.274336138189330, −7.859048288209660, −7.477357287362431, −6.921922360563598, −6.492937160692567, −6.064929899280196, −5.476633933439957, −5.045825954142554, −4.581681843191641, −3.810528421282767, −3.540724703301896, −2.955764816316354, −2.532202115450610, −2.033548316972204, −0.9510029128050573, −0.3646093908467576, 0.3646093908467576, 0.9510029128050573, 2.033548316972204, 2.532202115450610, 2.955764816316354, 3.540724703301896, 3.810528421282767, 4.581681843191641, 5.045825954142554, 5.476633933439957, 6.064929899280196, 6.492937160692567, 6.921922360563598, 7.477357287362431, 7.859048288209660, 8.274336138189330, 9.134352525493640, 9.395156429823512, 9.887909014864096, 10.33955510854648, 10.92337072642965, 11.20434034486187, 11.65942783423738, 12.46890474056297, 12.65601910793929

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