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Properties

Label 2-84e2-12.11-c1-0-56
Degree $2$
Conductor $7056$
Sign $0.995 + 0.0917i$
Analytic cond. $56.3424$
Root an. cond. $7.50616$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.732i·5-s + 4.24·11-s + 3.48·13-s + 2.73i·17-s − 1.79i·19-s + 0.656·23-s + 4.46·25-s − 2.44i·29-s − 6.69i·31-s + 3.46·37-s − 6.19i·41-s + 2.53i·43-s − 3.46·47-s + 1.41i·53-s + 3.10i·55-s + ⋯
L(s)  = 1  + 0.327i·5-s + 1.27·11-s + 0.966·13-s + 0.662i·17-s − 0.411i·19-s + 0.136·23-s + 0.892·25-s − 0.454i·29-s − 1.20i·31-s + 0.569·37-s − 0.967i·41-s + 0.386i·43-s − 0.505·47-s + 0.194i·53-s + 0.418i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.995 + 0.0917i$
Analytic conductor: \(56.3424\)
Root analytic conductor: \(7.50616\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7056} (4607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ 0.995 + 0.0917i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.462843565\)
\(L(\frac12)\) \(\approx\) \(2.462843565\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.732iT - 5T^{2} \)
11 \( 1 - 4.24T + 11T^{2} \)
13 \( 1 - 3.48T + 13T^{2} \)
17 \( 1 - 2.73iT - 17T^{2} \)
19 \( 1 + 1.79iT - 19T^{2} \)
23 \( 1 - 0.656T + 23T^{2} \)
29 \( 1 + 2.44iT - 29T^{2} \)
31 \( 1 + 6.69iT - 31T^{2} \)
37 \( 1 - 3.46T + 37T^{2} \)
41 \( 1 + 6.19iT - 41T^{2} \)
43 \( 1 - 2.53iT - 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 - 1.41iT - 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 + 3.20T + 61T^{2} \)
67 \( 1 + 0.928iT - 67T^{2} \)
71 \( 1 - 4.24T + 71T^{2} \)
73 \( 1 - 3.20T + 73T^{2} \)
79 \( 1 + 15.4iT - 79T^{2} \)
83 \( 1 + 9.46T + 83T^{2} \)
89 \( 1 + 7.66iT - 89T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−7.926414769801311491748815732740, −7.17834850056224694965860706186, −6.36416539615235274815599201992, −6.13013034061867539766742010858, −5.09775993067593480059723204096, −4.13223962944146151930898370493, −3.71055632267389660458548366882, −2.73922976459734576863842256226, −1.73116753820968324721342880818, −0.78718699878884859166343780270, 0.944977645826937015969475996629, 1.56963965087870548686477305623, 2.85467524626166116162138068512, 3.61448317139280480937999807100, 4.33227185063813113663631831123, 5.11736154374455117812567385315, 5.85443599259900842653448429534, 6.73375484701591606167561291270, 6.98303400688463452012224236255, 8.175655138413656504386454207910

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