Properties

Label 2-132e2-1.1-c1-0-55
Degree $2$
Conductor $17424$
Sign $-1$
Analytic cond. $139.131$
Root an. cond. $11.7953$
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  + 2·5-s + 2·13-s + 2·17-s − 4·19-s − 8·23-s − 25-s + 6·29-s − 8·31-s + 6·37-s − 6·41-s + 4·43-s − 7·49-s + 2·53-s + 4·59-s + 2·61-s + 4·65-s + 4·67-s + 8·71-s − 10·73-s − 8·79-s + 4·83-s + 4·85-s + 6·89-s − 8·95-s + 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.554·13-s + 0.485·17-s − 0.917·19-s − 1.66·23-s − 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.986·37-s − 0.937·41-s + 0.609·43-s − 49-s + 0.274·53-s + 0.520·59-s + 0.256·61-s + 0.496·65-s + 0.488·67-s + 0.949·71-s − 1.17·73-s − 0.900·79-s + 0.439·83-s + 0.433·85-s + 0.635·89-s − 0.820·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17424\)    =    \(2^{4} \cdot 3^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(139.131\)
Root analytic conductor: \(11.7953\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{17424} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 17424,\ (\ :1/2),\ -1)\)

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

−16.24688640800094, −15.65379297728913, −14.90336869465901, −14.38593959725412, −13.99230880412379, −13.32935866868449, −12.94968962861202, −12.25511467641868, −11.74497794529060, −11.00249003564484, −10.48590982347771, −9.872719359848364, −9.534217458881954, −8.694901520701798, −8.222404795939091, −7.612582757251573, −6.724723159923009, −6.202786469001169, −5.735595441825211, −5.076566067543094, −4.162527404982158, −3.672494877790323, −2.648356895289312, −2.000469175524955, −1.279461821856717, 0, 1.279461821856717, 2.000469175524955, 2.648356895289312, 3.672494877790323, 4.162527404982158, 5.076566067543094, 5.735595441825211, 6.202786469001169, 6.724723159923009, 7.612582757251573, 8.222404795939091, 8.694901520701798, 9.534217458881954, 9.872719359848364, 10.48590982347771, 11.00249003564484, 11.74497794529060, 12.25511467641868, 12.94968962861202, 13.32935866868449, 13.99230880412379, 14.38593959725412, 14.90336869465901, 15.65379297728913, 16.24688640800094

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