Properties

Label 2-22218-1.1-c1-0-27
Degree $2$
Conductor $22218$
Sign $-1$
Analytic cond. $177.411$
Root an. cond. $13.3195$
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 + 3-s + 4-s − 3·5-s + 6-s + 7-s + 8-s + 9-s − 3·10-s − 4·11-s + 12-s − 3·13-s + 14-s − 3·15-s + 16-s + 4·17-s + 18-s − 3·20-s + 21-s − 4·22-s + 24-s + 4·25-s − 3·26-s + 27-s + 28-s + 3·29-s − 3·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s − 1.20·11-s + 0.288·12-s − 0.832·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.670·20-s + 0.218·21-s − 0.852·22-s + 0.204·24-s + 4/5·25-s − 0.588·26-s + 0.192·27-s + 0.188·28-s + 0.557·29-s − 0.547·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22218\)    =    \(2 \cdot 3 \cdot 7 \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(177.411\)
Root analytic conductor: \(13.3195\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{22218} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 22218,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 - T \)
23 \( 1 \)
good5 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−15.77714870282122, −15.01580329576621, −14.72671118993419, −14.44320152020856, −13.60349378468572, −13.01230618404069, −12.62201489286784, −12.00930700009114, −11.62166128951789, −10.90081734147937, −10.47942782225066, −9.765863288832910, −9.106444251967951, −8.206800378307825, −7.857690297568590, −7.493261345667970, −7.002447800358863, −5.918324273486683, −5.389289905128725, −4.566208059344880, −4.290627161660511, −3.428315290114449, −2.868913293153329, −2.284314103281577, −1.138724637365272, 0, 1.138724637365272, 2.284314103281577, 2.868913293153329, 3.428315290114449, 4.290627161660511, 4.566208059344880, 5.389289905128725, 5.918324273486683, 7.002447800358863, 7.493261345667970, 7.857690297568590, 8.206800378307825, 9.106444251967951, 9.765863288832910, 10.47942782225066, 10.90081734147937, 11.62166128951789, 12.00930700009114, 12.62201489286784, 13.01230618404069, 13.60349378468572, 14.44320152020856, 14.72671118993419, 15.01580329576621, 15.77714870282122

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