Properties

Label 2-2816-176.109-c0-0-7
Degree $2$
Conductor $2816$
Sign $0.608 - 0.793i$
Analytic cond. $1.40536$
Root an. cond. $1.18548$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 1.22i)3-s + (0.366 − 0.366i)5-s + 1.99i·9-s + (0.707 − 0.707i)11-s + 0.896·15-s − 1.93i·23-s + 0.732i·25-s + (−1.22 + 1.22i)27-s + 1.93·31-s + 1.73·33-s + (−1.36 + 1.36i)37-s + (0.732 + 0.732i)45-s − 1.41·47-s − 49-s + (−1 + i)53-s + ⋯
L(s)  = 1  + (1.22 + 1.22i)3-s + (0.366 − 0.366i)5-s + 1.99i·9-s + (0.707 − 0.707i)11-s + 0.896·15-s − 1.93i·23-s + 0.732i·25-s + (−1.22 + 1.22i)27-s + 1.93·31-s + 1.73·33-s + (−1.36 + 1.36i)37-s + (0.732 + 0.732i)45-s − 1.41·47-s − 49-s + (−1 + i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2816\)    =    \(2^{8} \cdot 11\)
Sign: $0.608 - 0.793i$
Analytic conductor: \(1.40536\)
Root analytic conductor: \(1.18548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2816} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2816,\ (\ :0),\ 0.608 - 0.793i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.050275913\)
\(L(\frac12)\) \(\approx\) \(2.050275913\)
\(L(1)\) 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 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
5 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
7 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + 1.93iT - T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 - 1.93T + T^{2} \)
37 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 + (1 - i)T - iT^{2} \)
59 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
71 \( 1 - 0.517iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + 1.73iT - T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−9.002781884441916420916751764003, −8.508370075693138102024093879960, −8.049406698843647442055196754776, −6.77124091704688656129194040198, −6.00176877046593657219662111260, −4.75634922445108640242878944887, −4.51103089597039722215936226265, −3.32879057973694385233888515954, −2.85924811299139505397832001602, −1.56425053227395488905516020295, 1.42138190167877431712540879122, 2.07025216148177765683295256549, 3.05155436124470530409693150382, 3.77290569029853233861513881163, 4.97693544533763346467479143770, 6.22989200748214631483023188146, 6.68685957402726641829060987560, 7.42653345287502132061910886407, 8.029029031845814682780002896971, 8.741934597793421745482371816990

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