Properties

Label 2-3328-8.5-c1-0-0
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  − 3.21i·3-s + 1.62i·5-s − 3.82·7-s − 7.31·9-s − 6.08i·11-s − i·13-s + 5.21·15-s − 4.80·17-s + 0.144i·19-s + 12.2i·21-s − 2.22·23-s + 2.36·25-s + 13.8i·27-s + 0.228i·29-s + 6.33·31-s + ⋯
L(s)  = 1  − 1.85i·3-s + 0.725i·5-s − 1.44·7-s − 2.43·9-s − 1.83i·11-s − 0.277i·13-s + 1.34·15-s − 1.16·17-s + 0.0332i·19-s + 2.67i·21-s − 0.464·23-s + 0.473·25-s + 2.66i·27-s + 0.0424i·29-s + 1.13·31-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

\(L(1)\) \(\approx\) \(0.1154957013\)
\(L(\frac12)\) \(\approx\) \(0.1154957013\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.21iT - 3T^{2} \)
5 \( 1 - 1.62iT - 5T^{2} \)
7 \( 1 + 3.82T + 7T^{2} \)
11 \( 1 + 6.08iT - 11T^{2} \)
17 \( 1 + 4.80T + 17T^{2} \)
19 \( 1 - 0.144iT - 19T^{2} \)
23 \( 1 + 2.22T + 23T^{2} \)
29 \( 1 - 0.228iT - 29T^{2} \)
31 \( 1 - 6.33T + 31T^{2} \)
37 \( 1 - 2.10iT - 37T^{2} \)
41 \( 1 + 7.47T + 41T^{2} \)
43 \( 1 + 0.518iT - 43T^{2} \)
47 \( 1 + 12.9T + 47T^{2} \)
53 \( 1 + 3.47iT - 53T^{2} \)
59 \( 1 + 6.08iT - 59T^{2} \)
61 \( 1 - 5.37iT - 61T^{2} \)
67 \( 1 - 6.63iT - 67T^{2} \)
71 \( 1 - 14.7T + 71T^{2} \)
73 \( 1 - 3.00T + 73T^{2} \)
79 \( 1 - 16.1T + 79T^{2} \)
83 \( 1 - 9.59iT - 83T^{2} \)
89 \( 1 + 0.780T + 89T^{2} \)
97 \( 1 - 1.44T + 97T^{2} \)
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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

−8.362598929229546759412002532670, −8.089980451871442225378515917560, −6.86047268304954805816024735850, −6.55327032170873595468959119002, −6.23159802914653286750949958484, −5.29160219395433718971492995748, −3.50845591651141003364637748950, −3.02586266114241951971117475008, −2.27285277076652398211647787554, −0.870895587628678116902094873383, 0.04307868830021876206668877779, 2.16698898230070645161769671204, 3.18408741199394282793413355986, 4.01472699613537743461886779079, 4.69328279227591640349542677466, 5.06070435761369033391678906373, 6.28771152359860851656350985242, 6.79027346828482111236260582084, 8.067380895137014818360301964135, 8.925836637999734103501539321781

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