Properties

Label 2-2816-88.43-c1-0-34
Degree $2$
Conductor $2816$
Sign $0.989 - 0.145i$
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  − 3.06·3-s − 1.45i·5-s − 1.31·7-s + 6.39·9-s + (1.97 − 2.66i)11-s + 6.20·13-s + 4.45i·15-s + 7.25i·17-s − 0.642i·19-s + 4.02·21-s − 3.77i·23-s + 2.89·25-s − 10.3·27-s − 7.45·29-s + 3.50i·31-s + ⋯
L(s)  = 1  − 1.76·3-s − 0.649i·5-s − 0.496·7-s + 2.13·9-s + (0.596 − 0.802i)11-s + 1.71·13-s + 1.14i·15-s + 1.75i·17-s − 0.147i·19-s + 0.879·21-s − 0.787i·23-s + 0.578·25-s − 1.99·27-s − 1.38·29-s + 0.629i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2816\)    =    \(2^{8} \cdot 11\)
Sign: $0.989 - 0.145i$
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: \((2,\ 2816,\ (\ :1/2),\ 0.989 - 0.145i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9294410143\)
\(L(\frac12)\) \(\approx\) \(0.9294410143\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.97 + 2.66i)T \)
good3 \( 1 + 3.06T + 3T^{2} \)
5 \( 1 + 1.45iT - 5T^{2} \)
7 \( 1 + 1.31T + 7T^{2} \)
13 \( 1 - 6.20T + 13T^{2} \)
17 \( 1 - 7.25iT - 17T^{2} \)
19 \( 1 + 0.642iT - 19T^{2} \)
23 \( 1 + 3.77iT - 23T^{2} \)
29 \( 1 + 7.45T + 29T^{2} \)
31 \( 1 - 3.50iT - 31T^{2} \)
37 \( 1 - 4.81iT - 37T^{2} \)
41 \( 1 - 0.623iT - 41T^{2} \)
43 \( 1 + 3.07iT - 43T^{2} \)
47 \( 1 - 8.82iT - 47T^{2} \)
53 \( 1 + 10.7iT - 53T^{2} \)
59 \( 1 - 3.33T + 59T^{2} \)
61 \( 1 - 1.78T + 61T^{2} \)
67 \( 1 - 4.81T + 67T^{2} \)
71 \( 1 - 13.9iT - 71T^{2} \)
73 \( 1 - 4.76iT - 73T^{2} \)
79 \( 1 - 2.24T + 79T^{2} \)
83 \( 1 - 12.5iT - 83T^{2} \)
89 \( 1 - 3.89T + 89T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−8.677113078121700452917440059559, −8.280182252513553254798313310277, −6.85412206350291388834721104223, −6.32487191353895802928405365388, −5.88439730082618348527394170679, −5.13464604661901744828599311248, −4.11824660065329129923971592253, −3.54593591664275355936109802816, −1.53560431977219328759937035292, −0.824247315950556099086786685048, 0.58651276759158743190534046189, 1.76223019130208636370269285904, 3.30336341595986716106124254174, 4.12775451093731739735096845748, 5.07204733842769169292230472590, 5.82584881353247535785461890192, 6.42533837304830335997779775906, 7.04276472644920780032399648942, 7.60584452019434767857059157844, 9.120194824095331802815261188194

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