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Properties

Label 2-3328-8.5-c1-0-7
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  + 2.56i·3-s − 0.561i·5-s − 2.56·7-s − 3.56·9-s + 5.12i·11-s − i·13-s + 1.43·15-s + 5.68·17-s + 5.12i·19-s − 6.56i·21-s − 8·23-s + 4.68·25-s − 1.43i·27-s + 2i·29-s − 4·31-s + ⋯
L(s)  = 1  + 1.47i·3-s − 0.251i·5-s − 0.968·7-s − 1.18·9-s + 1.54i·11-s − 0.277i·13-s + 0.371·15-s + 1.37·17-s + 1.17i·19-s − 1.43i·21-s − 1.66·23-s + 0.936·25-s − 0.276i·27-s + 0.371i·29-s − 0.718·31-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

\(L(1)\) \(\approx\) \(0.7835640185\)
\(L(\frac12)\) \(\approx\) \(0.7835640185\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.56iT - 3T^{2} \)
5 \( 1 + 0.561iT - 5T^{2} \)
7 \( 1 + 2.56T + 7T^{2} \)
11 \( 1 - 5.12iT - 11T^{2} \)
17 \( 1 - 5.68T + 17T^{2} \)
19 \( 1 - 5.12iT - 19T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 9.68iT - 37T^{2} \)
41 \( 1 - 3.12T + 41T^{2} \)
43 \( 1 + 5.43iT - 43T^{2} \)
47 \( 1 - 0.315T + 47T^{2} \)
53 \( 1 - 3.12iT - 53T^{2} \)
59 \( 1 + 5.12iT - 59T^{2} \)
61 \( 1 + 11.1iT - 61T^{2} \)
67 \( 1 + 5.12iT - 67T^{2} \)
71 \( 1 + 7.68T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 2.24iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 8.24T + 97T^{2} \)
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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

−9.523851947255177860933762197361, −8.441881805076329759352560960832, −7.73284657144993139629011579885, −6.80738793910067143239010539490, −5.86404865058936746020254602235, −5.20145051813619632218506078443, −4.40616568112339884155506986859, −3.69630499746750695475512791868, −3.04373271962120945252096142383, −1.66829786522083326573348504962, 0.25256114060626209809722592994, 1.18291512268303474188976818208, 2.48039266927862469762676284510, 3.11723749653298498394034588845, 4.04594554369206501566350136854, 5.62769958454832672365464673097, 5.96216179932687821969944797932, 6.72140787330322599246580326687, 7.34421718115972697228783277324, 8.021957933529069719060475979078

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