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Properties

Label 2-84e2-1.1-c1-0-31
Degree $2$
Conductor $7056$
Sign $1$
Analytic cond. $56.3424$
Root an. cond. $7.50616$
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  + 4·11-s + 8·23-s − 5·25-s − 2·29-s − 6·37-s + 12·43-s + 10·53-s − 4·67-s + 16·71-s − 8·79-s − 20·107-s + 18·109-s − 2·113-s + ⋯
L(s)  = 1  + 1.20·11-s + 1.66·23-s − 25-s − 0.371·29-s − 0.986·37-s + 1.82·43-s + 1.37·53-s − 0.488·67-s + 1.89·71-s − 0.900·79-s − 1.93·107-s + 1.72·109-s − 0.188·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(56.3424\)
Root analytic conductor: \(7.50616\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.232396067\)
\(L(\frac12)\) \(\approx\) \(2.232396067\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−7.88015248379246614186817321181, −7.15577725211446526254897397286, −6.65412369656665464490018383618, −5.82069989900162637224273703456, −5.17870506317096288141302501922, −4.22202893804670973052984956023, −3.68314720218148238252015815535, −2.73945479676999935774549485152, −1.73836198313686996977969819572, −0.789590281538587110436847636677, 0.789590281538587110436847636677, 1.73836198313686996977969819572, 2.73945479676999935774549485152, 3.68314720218148238252015815535, 4.22202893804670973052984956023, 5.17870506317096288141302501922, 5.82069989900162637224273703456, 6.65412369656665464490018383618, 7.15577725211446526254897397286, 7.88015248379246614186817321181

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