Properties

Label 2-3328-8.5-c1-0-13
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  + 0.378i·3-s − 4.27i·5-s − 4.43·7-s + 2.85·9-s + 2.38i·11-s − i·13-s + 1.62·15-s − 3.51·17-s + 7.74i·19-s − 1.67i·21-s − 1.35·23-s − 13.2·25-s + 2.21i·27-s − 0.643i·29-s + 7.63·31-s + ⋯
L(s)  = 1  + 0.218i·3-s − 1.91i·5-s − 1.67·7-s + 0.952·9-s + 0.720i·11-s − 0.277i·13-s + 0.418·15-s − 0.853·17-s + 1.77i·19-s − 0.366i·21-s − 0.282·23-s − 2.65·25-s + 0.427i·27-s − 0.119i·29-s + 1.37·31-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

\(L(1)\) \(\approx\) \(1.027064661\)
\(L(\frac12)\) \(\approx\) \(1.027064661\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.378iT - 3T^{2} \)
5 \( 1 + 4.27iT - 5T^{2} \)
7 \( 1 + 4.43T + 7T^{2} \)
11 \( 1 - 2.38iT - 11T^{2} \)
17 \( 1 + 3.51T + 17T^{2} \)
19 \( 1 - 7.74iT - 19T^{2} \)
23 \( 1 + 1.35T + 23T^{2} \)
29 \( 1 + 0.643iT - 29T^{2} \)
31 \( 1 - 7.63T + 31T^{2} \)
37 \( 1 - 5.09iT - 37T^{2} \)
41 \( 1 - 5.19T + 41T^{2} \)
43 \( 1 + 1.19iT - 43T^{2} \)
47 \( 1 + 2.09T + 47T^{2} \)
53 \( 1 - 9.19iT - 53T^{2} \)
59 \( 1 - 2.38iT - 59T^{2} \)
61 \( 1 - 3.68iT - 61T^{2} \)
67 \( 1 + 9.36iT - 67T^{2} \)
71 \( 1 + 9.96T + 71T^{2} \)
73 \( 1 + 6.26T + 73T^{2} \)
79 \( 1 + 0.777T + 79T^{2} \)
83 \( 1 + 7.76iT - 83T^{2} \)
89 \( 1 - 7.61T + 89T^{2} \)
97 \( 1 - 8.97T + 97T^{2} \)
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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

−8.871360157880493072271808713352, −8.060641148309221090515481355143, −7.32054183724997997815405416212, −6.27795787648706065473438432541, −5.79042697900178277781983945179, −4.60160208498083037689359299135, −4.31207682148588630113172318311, −3.36481251935627345385047672417, −1.96928012324912441347537904161, −0.953890543445829871237657063031, 0.37299904123041270897657546820, 2.28828055602796118317971124724, 2.89262583765992781161820552014, 3.59891886515445885604570774132, 4.47054834637818302299463955959, 5.97848553048962193602212628212, 6.45592192215257032653758746993, 6.96511093842057064004164154480, 7.36078633563857726335928623431, 8.576595661341034799365642629974

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