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Properties

Label 2-3328-8.5-c1-0-10
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  + 2.23i·3-s − 3i·5-s − 2.23·7-s − 2.00·9-s + 4.47i·11-s − i·13-s + 6.70·15-s − 3·17-s − 4.47i·19-s − 5.00i·21-s + 8.94·23-s − 4·25-s + 2.23i·27-s + 10i·29-s − 10.0·33-s + ⋯
L(s)  = 1  + 1.29i·3-s − 1.34i·5-s − 0.845·7-s − 0.666·9-s + 1.34i·11-s − 0.277i·13-s + 1.73·15-s − 0.727·17-s − 1.02i·19-s − 1.09i·21-s + 1.86·23-s − 0.800·25-s + 0.430i·27-s + 1.85i·29-s − 1.74·33-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

\(L(1)\) \(\approx\) \(1.047023399\)
\(L(\frac12)\) \(\approx\) \(1.047023399\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23iT - 3T^{2} \)
5 \( 1 + 3iT - 5T^{2} \)
7 \( 1 + 2.23T + 7T^{2} \)
11 \( 1 - 4.47iT - 11T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + 4.47iT - 19T^{2} \)
23 \( 1 - 8.94T + 23T^{2} \)
29 \( 1 - 10iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 3iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 6.70iT - 43T^{2} \)
47 \( 1 - 2.23T + 47T^{2} \)
53 \( 1 + 4iT - 53T^{2} \)
59 \( 1 - 4.47iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 13.4iT - 67T^{2} \)
71 \( 1 + 6.70T + 71T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 + 8.94T + 79T^{2} \)
83 \( 1 - 17.8iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−9.053925337611489955523338613967, −8.669344238771729829735942059260, −7.23020843772506476284103900136, −6.84866079041310868796611732616, −5.45387803173768776976184704064, −4.93670705825449601930891939898, −4.47870854996578677207664046055, −3.59811192331837045214888222113, −2.60814400776212100639895005951, −1.16725832193935022155067138059, 0.34352469053775353781739827941, 1.65138137018106984412920096306, 2.85817313257320118487583343621, 3.13000360775533887724957644628, 4.33468644254046648056491850628, 5.86379107399020928099026297478, 6.26603900581626992646417359105, 6.77980492793573926877724113882, 7.45564683007980874631160375630, 8.137529171714767835562183514671

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