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Properties

Label 2-333270-1.1-c1-0-124
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 $1$

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 + 4·13-s − 14-s + 16-s − 3·17-s + 5·19-s + 20-s − 4·22-s + 25-s − 4·26-s + 28-s + 10·29-s − 2·31-s − 32-s + 3·34-s + 35-s + 3·37-s − 5·38-s − 40-s − 4·41-s − 8·43-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 + 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 1.14·19-s + 0.223·20-s − 0.852·22-s + 1/5·25-s − 0.784·26-s + 0.188·28-s + 1.85·29-s − 0.359·31-s − 0.176·32-s + 0.514·34-s + 0.169·35-s + 0.493·37-s − 0.811·38-s − 0.158·40-s − 0.624·41-s − 1.21·43-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: \(1\)
Selberg data: \((2,\ 333270,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 13 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 15 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−12.83404044943995, −12.15533807899692, −11.74787878855906, −11.45722936210855, −11.05559700816609, −10.49277304082904, −9.927354902276703, −9.755693003346012, −8.995132974727379, −8.821879895172003, −8.299357469755655, −7.950674416337473, −7.172426567431282, −6.793283862841473, −6.349822406983750, −6.053543119850403, −5.292707713705212, −4.855073247356190, −4.247999227086280, −3.627893793978673, −3.152651704158054, −2.560656679377725, −1.790941542690814, −1.308217384183249, −1.016255481805045, 0, 1.016255481805045, 1.308217384183249, 1.790941542690814, 2.560656679377725, 3.152651704158054, 3.627893793978673, 4.247999227086280, 4.855073247356190, 5.292707713705212, 6.053543119850403, 6.349822406983750, 6.793283862841473, 7.172426567431282, 7.950674416337473, 8.299357469755655, 8.821879895172003, 8.995132974727379, 9.755693003346012, 9.927354902276703, 10.49277304082904, 11.05559700816609, 11.45722936210855, 11.74787878855906, 12.15533807899692, 12.83404044943995

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