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Properties

Label 2-2816-8.5-c1-0-69
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s + 3i·5-s + 2·7-s − 6·9-s + i·11-s + 9·15-s − 6·17-s + 4i·19-s − 6i·21-s − 23-s − 4·25-s + 9i·27-s − 8i·29-s − 7·31-s + 3·33-s + ⋯
L(s)  = 1  − 1.73i·3-s + 1.34i·5-s + 0.755·7-s − 2·9-s + 0.301i·11-s + 2.32·15-s − 1.45·17-s + 0.917i·19-s − 1.30i·21-s − 0.208·23-s − 0.800·25-s + 1.73i·27-s − 1.48i·29-s − 1.25·31-s + 0.522·33-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: $\chi_{2816} (1409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2816,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good3 \( 1 + 3iT - 3T^{2} \)
5 \( 1 - 3iT - 5T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 + 8iT - 29T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 - iT - 37T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 - iT - 59T^{2} \)
61 \( 1 - 4iT - 61T^{2} \)
67 \( 1 + 5iT - 67T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 + 16T + 73T^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + 2iT - 83T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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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

−8.075456953892517746950652567850, −7.44751026407531014052209263976, −6.88144998029354947213877375892, −6.31224205612578957695297063512, −5.57709912063237667297711539181, −4.30304610681840589751701117378, −3.14509667751580588413201927457, −2.16708329662981525734307957068, −1.72030039433798048216719571572, 0, 1.62688877612980138028391237014, 3.00373627367067330663664135567, 4.01654075476779087146979310164, 4.76600576333683468982658363119, 4.96481515543037221036661321636, 5.81800360841881394382967765084, 7.00211539241103162789413497654, 8.237924353526041549202947406786, 8.751521086958843013245678223767

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