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Properties

Label 2-2816-8.5-c1-0-54
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  + 3i·3-s − i·5-s − 6·9-s + i·11-s − 6i·13-s + 3·15-s − 4·17-s + 6i·19-s − 3·23-s + 4·25-s − 9i·27-s − 4i·29-s − 9·31-s − 3·33-s − 7i·37-s + ⋯
L(s)  = 1  + 1.73i·3-s − 0.447i·5-s − 2·9-s + 0.301i·11-s − 1.66i·13-s + 0.774·15-s − 0.970·17-s + 1.37i·19-s − 0.625·23-s + 0.800·25-s − 1.73i·27-s − 0.742i·29-s − 1.61·31-s − 0.522·33-s − 1.15i·37-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

\(L(1)\) \(\approx\) \(0.8094066942\)
\(L(\frac12)\) \(\approx\) \(0.8094066942\)
\(L(\frac{3}{2})\) 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 \( 1 + iT - 5T^{2} \)
7 \( 1 + 7T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 + 9T + 31T^{2} \)
37 \( 1 + 7iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 + 9iT - 59T^{2} \)
61 \( 1 - 8iT - 61T^{2} \)
67 \( 1 + 15iT - 67T^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 5T + 89T^{2} \)
97 \( 1 + 3T + 97T^{2} \)
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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

−8.828166532879313063266760358486, −8.203231858226989815633773288995, −7.37429782400931558499658528270, −5.95923148958372934368105614052, −5.51890285313055267382068267363, −4.73989642467808185576099402811, −3.94386681295991909761339123533, −3.35202094195077612571948163769, −2.15859539392913729547004501638, −0.26483538563779084555808931266, 1.20056245821313530318904163647, 2.16227545382332574653939063753, 2.82752987980947634028591631281, 4.11146394682894729592621045229, 5.16248964374009870609883321588, 6.25325784827471714775020220923, 6.77404503145119496374355093687, 7.11198342021762712135026152189, 7.970929161623730609973162627095, 8.965815256709578061755701176364

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