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Properties

Label 2-2816-88.43-c1-0-90
Degree $2$
Conductor $2816$
Sign $-0.707 + 0.707i$
Analytic cond. $22.4858$
Root an. cond. $4.74192$
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  + 2.52·3-s − 4.37i·5-s + 3.37·9-s − 3.31·11-s − 11.0i·15-s − 9.45i·23-s − 14.1·25-s + 0.939·27-s − 0.644i·31-s − 8.37·33-s + 5.11i·37-s − 14.7i·45-s − 6.63i·47-s − 7·49-s + 6i·53-s + ⋯
L(s)  = 1  + 1.45·3-s − 1.95i·5-s + 1.12·9-s − 1.00·11-s − 2.84i·15-s − 1.97i·23-s − 2.82·25-s + 0.180·27-s − 0.115i·31-s − 1.45·33-s + 0.841i·37-s − 2.19i·45-s − 0.967i·47-s − 49-s + 0.824i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2816\)    =    \(2^{8} \cdot 11\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(22.4858\)
Root analytic conductor: \(4.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2816} (1407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2816,\ (\ :1/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.316743534\)
\(L(\frac12)\) \(\approx\) \(2.316743534\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + 3.31T \)
good3 \( 1 - 2.52T + 3T^{2} \)
5 \( 1 + 4.37iT - 5T^{2} \)
7 \( 1 + 7T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 9.45iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 0.644iT - 31T^{2} \)
37 \( 1 - 5.11iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 6.63iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 6.28T + 67T^{2} \)
71 \( 1 + 5.69iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 9.86T + 89T^{2} \)
97 \( 1 - 17.1T + 97T^{2} \)
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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

−8.433321326212557279164027700340, −8.193656525410058352540610942110, −7.37388685684881963943492452278, −6.15186072359997352833266933892, −5.10962063406740102616929201454, −4.59788704127813929649916470654, −3.73282456915206013283468912507, −2.63020972675035005745179977573, −1.82935358217118382244111193757, −0.55454368284356656594351880863, 1.96430246124001863987949287617, 2.60557543383754983044638099584, 3.35820371354280882275928517754, 3.80018348768993524784330876809, 5.27788289420606788957791292399, 6.19743368135999328847256854077, 7.12055203815994487007197774686, 7.62102137311057910031891365580, 8.073747961959602077515745810537, 9.133294239876823339550416770711

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