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Properties

Label 2-330e2-1.1-c1-0-35
Degree $2$
Conductor $108900$
Sign $1$
Analytic cond. $869.570$
Root an. cond. $29.4884$
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  + 2·7-s + 2·13-s − 2·19-s + 8·31-s − 2·37-s + 2·43-s − 3·49-s + 6·53-s + 12·59-s − 2·61-s + 4·67-s + 2·73-s + 10·79-s + 12·83-s + 6·89-s + 4·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.755·7-s + 0.554·13-s − 0.458·19-s + 1.43·31-s − 0.328·37-s + 0.304·43-s − 3/7·49-s + 0.824·53-s + 1.56·59-s − 0.256·61-s + 0.488·67-s + 0.234·73-s + 1.12·79-s + 1.31·83-s + 0.635·89-s + 0.419·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(869.570\)
Root analytic conductor: \(29.4884\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 108900,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.191357091\)
\(L(\frac12)\) \(\approx\) \(3.191357091\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.71163018082844, −13.25809923064762, −12.68608656878328, −12.14887734662012, −11.72739058742553, −11.18714378217071, −10.84642208413090, −10.22299260721976, −9.849780317134017, −9.164694959947978, −8.576392635892302, −8.316914520145896, −7.757943453508067, −7.198087778758349, −6.541652003524602, −6.183112417721445, −5.483939474839659, −4.953145029514745, −4.471490291912491, −3.841331492893728, −3.313039813726561, −2.464754097303663, −2.017468930191404, −1.197066666872102, −0.6023488678225135, 0.6023488678225135, 1.197066666872102, 2.017468930191404, 2.464754097303663, 3.313039813726561, 3.841331492893728, 4.471490291912491, 4.953145029514745, 5.483939474839659, 6.183112417721445, 6.541652003524602, 7.198087778758349, 7.757943453508067, 8.316914520145896, 8.576392635892302, 9.164694959947978, 9.849780317134017, 10.22299260721976, 10.84642208413090, 11.18714378217071, 11.72739058742553, 12.14887734662012, 12.68608656878328, 13.25809923064762, 13.71163018082844

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