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Properties

Label 2-333270-1.1-c1-0-0
Degree $2$
Conductor $333270$
Sign $1$
Analytic cond. $2661.17$
Root an. cond. $51.5865$
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 − 7-s + 8-s − 10-s − 4·11-s − 2·13-s − 14-s + 16-s + 4·19-s − 20-s − 4·22-s + 25-s − 2·26-s − 28-s − 8·29-s − 8·31-s + 32-s + 35-s − 6·37-s + 4·38-s − 40-s + 8·41-s + 8·43-s − 4·44-s + 49-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s − 1.20·11-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.917·19-s − 0.223·20-s − 0.852·22-s + 1/5·25-s − 0.392·26-s − 0.188·28-s − 1.48·29-s − 1.43·31-s + 0.176·32-s + 0.169·35-s − 0.986·37-s + 0.648·38-s − 0.158·40-s + 1.24·41-s + 1.21·43-s − 0.603·44-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(333270\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(2661.17\)
Root analytic conductor: \(51.5865\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 333270,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1564746639\)
\(L(\frac12)\) \(\approx\) \(0.1564746639\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.68672913671682, −12.25101396720493, −11.80051722920467, −11.14835136565983, −10.87264093904253, −10.57294211454555, −9.792286692838394, −9.457687095899675, −9.069440541873634, −8.299956995622071, −7.798085270626643, −7.426604590094792, −7.164946304469844, −6.566216679806407, −5.746063381712137, −5.536142552241129, −5.196012812521984, −4.418615595283780, −4.077097235428986, −3.446506587193753, −2.928481837096469, −2.584966280746032, −1.830212167566351, −1.230642070090482, −0.08819463683546740, 0.08819463683546740, 1.230642070090482, 1.830212167566351, 2.584966280746032, 2.928481837096469, 3.446506587193753, 4.077097235428986, 4.418615595283780, 5.196012812521984, 5.536142552241129, 5.746063381712137, 6.566216679806407, 7.164946304469844, 7.426604590094792, 7.798085270626643, 8.299956995622071, 9.069440541873634, 9.457687095899675, 9.792286692838394, 10.57294211454555, 10.87264093904253, 11.14835136565983, 11.80051722920467, 12.25101396720493, 12.68672913671682

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