Properties

Label 2-84e2-28.27-c1-0-17
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  − 1.73i·5-s + 5.19i·11-s + 6.92i·13-s − 3.46i·17-s − 2·19-s − 6.92i·23-s + 2.00·25-s + 9·29-s − 31-s − 2·37-s + 3.46i·41-s + 3.46i·43-s − 9·53-s + 9·55-s − 3·59-s + ⋯
L(s)  = 1  − 0.774i·5-s + 1.56i·11-s + 1.92i·13-s − 0.840i·17-s − 0.458·19-s − 1.44i·23-s + 0.400·25-s + 1.67·29-s − 0.179·31-s − 0.328·37-s + 0.541i·41-s + 0.528i·43-s − 1.23·53-s + 1.21·55-s − 0.390·59-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

\(L(1)\) \(\approx\) \(1.306273504\)
\(L(\frac12)\) \(\approx\) \(1.306273504\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT - 5T^{2} \)
11 \( 1 - 5.19iT - 11T^{2} \)
13 \( 1 - 6.92iT - 13T^{2} \)
17 \( 1 + 3.46iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 6.92iT - 23T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 - 3.46iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + 3T + 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 6.92iT - 71T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 + 1.73iT - 79T^{2} \)
83 \( 1 + 15T + 83T^{2} \)
89 \( 1 + 10.3iT - 89T^{2} \)
97 \( 1 - 8.66iT - 97T^{2} \)
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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

−8.254762292639315656057556791450, −7.30278816188348665662173856129, −6.74611835301507404003353578648, −6.26622755948430460701194563576, −4.86473286169586115822361043206, −4.69441245099543341064498590501, −4.15667341144945807492761991799, −2.77891514688239212122792227168, −2.02649648965153145572762321785, −1.14616200313891886986538043202, 0.33646107441497139943307893810, 1.43190693171559464666531507176, 2.83971158969057842666909100251, 3.16289797841770483105960018290, 3.89668402377204552347919205273, 5.10917350673191009348646510949, 5.73265661698145247940489403921, 6.24799844820075492541690941678, 7.02009639931977635766901108503, 7.902561194151981113991450976962

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