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Properties

Label 2-66e2-1.1-c1-0-44
Degree $2$
Conductor $4356$
Sign $-1$
Analytic cond. $34.7828$
Root an. cond. $5.89769$
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  + 4·7-s − 2·13-s − 8·19-s − 5·25-s − 4·31-s − 10·37-s − 8·43-s + 9·49-s − 14·61-s − 16·67-s + 10·73-s + 4·79-s − 8·91-s + 14·97-s + 20·103-s − 2·109-s + ⋯
L(s)  = 1  + 1.51·7-s − 0.554·13-s − 1.83·19-s − 25-s − 0.718·31-s − 1.64·37-s − 1.21·43-s + 9/7·49-s − 1.79·61-s − 1.95·67-s + 1.17·73-s + 0.450·79-s − 0.838·91-s + 1.42·97-s + 1.97·103-s − 0.191·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4356\)    =    \(2^{2} \cdot 3^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(34.7828\)
Root analytic conductor: \(5.89769\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4356,\ (\ :1/2),\ -1)\)

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 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.010673460704295170352395083856, −7.42994097792857760856706842065, −6.57239462159071045396318968839, −5.72957130079092448905158380742, −4.89966072388805335574106632098, −4.40802673262821988065829894247, −3.44955454863415035311235849889, −2.13786510902797553101496444618, −1.67345259685519684395551686906, 0, 1.67345259685519684395551686906, 2.13786510902797553101496444618, 3.44955454863415035311235849889, 4.40802673262821988065829894247, 4.89966072388805335574106632098, 5.72957130079092448905158380742, 6.57239462159071045396318968839, 7.42994097792857760856706842065, 8.010673460704295170352395083856

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