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Properties

Label 2-84e2-28.27-c1-0-25
Degree $2$
Conductor $7056$
Sign $0.188 - 0.981i$
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  + 5.19i·13-s + 19-s + 5·25-s + 11·31-s − 11·37-s − 1.73i·43-s − 6.92i·61-s + 15.5i·67-s − 1.73i·73-s − 5.19i·79-s + 13.8i·97-s − 13·103-s − 19·109-s + ⋯
L(s)  = 1  + 1.44i·13-s + 0.229·19-s + 25-s + 1.97·31-s − 1.80·37-s − 0.264i·43-s − 0.887i·61-s + 1.90i·67-s − 0.202i·73-s − 0.584i·79-s + 1.40i·97-s − 1.28·103-s − 1.81·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.188 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.188 - 0.981i$
Analytic conductor: \(56.3424\)
Root analytic conductor: \(7.50616\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7056} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ 0.188 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.713671318\)
\(L(\frac12)\) \(\approx\) \(1.713671318\)
\(L(\frac{3}{2})\) 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 \( 1 - 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 5.19iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 11T + 31T^{2} \)
37 \( 1 + 11T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 1.73iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 6.92iT - 61T^{2} \)
67 \( 1 - 15.5iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 + 5.19iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−8.249416535819423769665807771719, −7.19628112851709556812531088172, −6.78831281187718995654389401246, −6.11873051863819844770428550406, −5.14637623569683469163390347620, −4.57168111924878068973262162250, −3.78971803175556855580617316606, −2.89482439717159791231926432902, −2.00332019508916376505888107558, −1.03365672588160536666719962832, 0.47466502773169828647330954501, 1.48305253095533316570950297007, 2.75389265934250115147210779291, 3.19416851684550124691527275008, 4.22579858963699140536395755106, 5.05219527495484853446663910657, 5.57520286773483361209965530303, 6.46194355014318870514094838768, 7.03879484629880095577861276117, 7.939080225461571127060879326265

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