Properties

Label 2-3328-8.5-c1-0-60
Degree $2$
Conductor $3328$
Sign $0.707 + 0.707i$
Analytic cond. $26.5742$
Root an. cond. $5.15501$
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  − 1.40i·3-s + 0.422i·5-s + 4.72·7-s + 1.02·9-s + 3.71i·11-s + i·13-s + 0.593·15-s + 2.39·17-s − 8.15i·19-s − 6.64i·21-s − 7.87·23-s + 4.82·25-s − 5.65i·27-s − 5.87i·29-s − 0.528·31-s + ⋯
L(s)  = 1  − 0.811i·3-s + 0.188i·5-s + 1.78·7-s + 0.340·9-s + 1.12i·11-s + 0.277i·13-s + 0.153·15-s + 0.579·17-s − 1.87i·19-s − 1.44i·21-s − 1.64·23-s + 0.964·25-s − 1.08i·27-s − 1.09i·29-s − 0.0948·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(26.5742\)
Root analytic conductor: \(5.15501\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (1665, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.561125428\)
\(L(\frac12)\) \(\approx\) \(2.561125428\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 - iT \)
good3 \( 1 + 1.40iT - 3T^{2} \)
5 \( 1 - 0.422iT - 5T^{2} \)
7 \( 1 - 4.72T + 7T^{2} \)
11 \( 1 - 3.71iT - 11T^{2} \)
17 \( 1 - 2.39T + 17T^{2} \)
19 \( 1 + 8.15iT - 19T^{2} \)
23 \( 1 + 7.87T + 23T^{2} \)
29 \( 1 + 5.87iT - 29T^{2} \)
31 \( 1 + 0.528T + 31T^{2} \)
37 \( 1 + 1.20iT - 37T^{2} \)
41 \( 1 + 9.03T + 41T^{2} \)
43 \( 1 - 2.19iT - 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 - 5.03iT - 53T^{2} \)
59 \( 1 - 3.71iT - 59T^{2} \)
61 \( 1 - 14.6iT - 61T^{2} \)
67 \( 1 + 6.47iT - 67T^{2} \)
71 \( 1 - 9.34T + 71T^{2} \)
73 \( 1 - 16.5T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 + 6.08iT - 83T^{2} \)
89 \( 1 + 8.63T + 89T^{2} \)
97 \( 1 + 0.755T + 97T^{2} \)
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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

−8.306564178013451443421555528542, −7.67718672670557181778377676312, −7.20704420244364566118443879778, −6.51399797432070393400105396558, −5.41923581476042903759689377695, −4.63027703540804491383635234687, −4.13541472816559981446429820591, −2.45666883964982631957411232088, −1.93354509112435761981041864430, −0.946753958689714618805258522391, 1.14428563266913065714111824810, 1.98738259066472749569406486465, 3.55108743279336654173889089440, 3.93347679378604215658125091056, 5.16159006767093336425905060943, 5.25051201461806897086922457191, 6.33074527572323264144617288238, 7.48886903518177157274799787981, 8.286353250021058783444351219142, 8.389553443702200149183290275819

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