Properties

Label 2-2816-8.5-c1-0-61
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  i·3-s + i·5-s − 2·7-s + 2·9-s i·11-s − 4i·13-s + 15-s − 2·17-s + 2i·21-s − 23-s + 4·25-s − 5i·27-s − 7·31-s − 33-s − 2i·35-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.447i·5-s − 0.755·7-s + 0.666·9-s − 0.301i·11-s − 1.10i·13-s + 0.258·15-s − 0.485·17-s + 0.436i·21-s − 0.208·23-s + 0.800·25-s − 0.962i·27-s − 1.25·31-s − 0.174·33-s − 0.338i·35-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: \(0\)
Selberg data: \((2,\ 2816,\ (\ :1/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.031539122\)
\(L(\frac12)\) \(\approx\) \(1.031539122\)
\(L(\frac{3}{2})\) 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 \( 1 + iT - 3T^{2} \)
5 \( 1 - iT - 5T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 - 3iT - 37T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 5iT - 59T^{2} \)
61 \( 1 + 12iT - 61T^{2} \)
67 \( 1 + 7iT - 67T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
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

−8.302527391626652276714107347508, −7.75164703125310646869963697921, −6.86060363043089862802918818442, −6.45533917905948251157907123640, −5.57615488775750965738337897656, −4.57760761337970861208606619946, −3.48600677096762046970341889300, −2.82555373629782158111136139760, −1.65194144038665596709273579850, −0.33422137679735750404242182948, 1.36747329116460545625661621007, 2.50457592797864243802896726666, 3.74543333508320484426818847308, 4.28360659033283494199441510909, 5.05663068126408195084243703201, 6.01538940583552023620483867352, 6.91679271442840710235842315596, 7.36375199317858547820177410741, 8.595995873790237691757765938325, 9.227180407775387332793625990020

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