Properties

Label 2-2816-88.43-c1-0-62
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  + 3.31·3-s + 3i·5-s + 8·9-s + 3.31·11-s + 9.94i·15-s − 3.31i·23-s − 4·25-s + 16.5·27-s − 9.94i·31-s + 11·33-s + 7i·37-s + 24i·45-s + 6.63i·47-s − 7·49-s + 6i·53-s + ⋯
L(s)  = 1  + 1.91·3-s + 1.34i·5-s + 2.66·9-s + 1.00·11-s + 2.56i·15-s − 0.691i·23-s − 0.800·25-s + 3.19·27-s − 1.78i·31-s + 1.91·33-s + 1.15i·37-s + 3.57i·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

\(L(1)\) \(\approx\) \(4.164253662\)
\(L(\frac12)\) \(\approx\) \(4.164253662\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.31T + 3T^{2} \)
5 \( 1 - 3iT - 5T^{2} \)
7 \( 1 + 7T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 3.31iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 9.94iT - 31T^{2} \)
37 \( 1 - 7iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 6.63iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 3.31T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 9.94T + 67T^{2} \)
71 \( 1 + 16.5iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + 17T + 97T^{2} \)
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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

−9.004101929207985475408220972290, −7.973261088010321470903721658248, −7.63748514795801563898215215484, −6.69706700067729986285344019398, −6.28112950243654580729310155764, −4.56400322265379211794966593772, −3.86754329358391859924914604225, −3.08465445247433503966741023318, −2.52183389423270904862779536715, −1.52562431655728173306011074383, 1.21986679994038940261050732527, 1.86137532145363574658900962186, 3.09277439014469618477807125665, 3.84820595234986949428889067284, 4.51761137104174719945196121106, 5.40332961934681688588666639330, 6.74073489459835952980295578983, 7.37106032732570703157523247940, 8.346137031523468576610632620350, 8.594219145927245831589693873961

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