Properties

Label 2-259182-1.1-c1-0-13
Degree $2$
Conductor $259182$
Sign $1$
Analytic cond. $2069.57$
Root an. cond. $45.4926$
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 − 3·5-s + 7-s + 8-s − 3·10-s + 2·13-s + 14-s + 16-s − 17-s + 7·19-s − 3·20-s − 23-s + 4·25-s + 2·26-s + 28-s + 8·29-s − 9·31-s + 32-s − 34-s − 3·35-s − 5·37-s + 7·38-s − 3·40-s + 2·41-s − 8·43-s − 46-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s + 0.353·8-s − 0.948·10-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 1.60·19-s − 0.670·20-s − 0.208·23-s + 4/5·25-s + 0.392·26-s + 0.188·28-s + 1.48·29-s − 1.61·31-s + 0.176·32-s − 0.171·34-s − 0.507·35-s − 0.821·37-s + 1.13·38-s − 0.474·40-s + 0.312·41-s − 1.21·43-s − 0.147·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(259182\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(2069.57\)
Root analytic conductor: \(45.4926\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 259182,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.270467823\)
\(L(\frac12)\) \(\approx\) \(2.270467823\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
17 \( 1 + T \)
good5 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−12.61383553582570, −12.26217275280497, −12.03067243058453, −11.47167967327613, −11.11468852936730, −10.76751416812274, −10.19930818969256, −9.565573632763120, −9.017934980519969, −8.537279679075016, −7.923437366138696, −7.704710015177908, −7.222429513523067, −6.645941436803022, −6.217521895047461, −5.493764948663509, −5.022828830144656, −4.681387577758908, −3.984932115362002, −3.618627370426701, −3.160433530081627, −2.652098599138663, −1.658888015450459, −1.296319491168591, −0.3664979693578568, 0.3664979693578568, 1.296319491168591, 1.658888015450459, 2.652098599138663, 3.160433530081627, 3.618627370426701, 3.984932115362002, 4.681387577758908, 5.022828830144656, 5.493764948663509, 6.217521895047461, 6.645941436803022, 7.222429513523067, 7.704710015177908, 7.923437366138696, 8.537279679075016, 9.017934980519969, 9.565573632763120, 10.19930818969256, 10.76751416812274, 11.11468852936730, 11.47167967327613, 12.03067243058453, 12.26217275280497, 12.61383553582570

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