Properties

Label 2-3328-8.5-c1-0-48
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  − 0.738i·3-s + 3.70i·5-s + 1.43·7-s + 2.45·9-s + 2.99i·11-s − i·13-s + 2.73·15-s + 2.23·17-s − 1.24i·19-s − 1.05i·21-s + 8.24·23-s − 8.75·25-s − 4.02i·27-s − 10.2i·29-s + 10.4·31-s + ⋯
L(s)  = 1  − 0.426i·3-s + 1.65i·5-s + 0.541·7-s + 0.818·9-s + 0.902i·11-s − 0.277i·13-s + 0.707·15-s + 0.541·17-s − 0.286i·19-s − 0.230i·21-s + 1.71·23-s − 1.75·25-s − 0.775i·27-s − 1.90i·29-s + 1.88·31-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

\(L(1)\) \(\approx\) \(2.381124338\)
\(L(\frac12)\) \(\approx\) \(2.381124338\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.738iT - 3T^{2} \)
5 \( 1 - 3.70iT - 5T^{2} \)
7 \( 1 - 1.43T + 7T^{2} \)
11 \( 1 - 2.99iT - 11T^{2} \)
17 \( 1 - 2.23T + 17T^{2} \)
19 \( 1 + 1.24iT - 19T^{2} \)
23 \( 1 - 8.24T + 23T^{2} \)
29 \( 1 + 10.2iT - 29T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 - 6.93iT - 37T^{2} \)
41 \( 1 + 1.17T + 41T^{2} \)
43 \( 1 + 9.90iT - 43T^{2} \)
47 \( 1 - 9.11T + 47T^{2} \)
53 \( 1 - 2.82iT - 53T^{2} \)
59 \( 1 - 2.99iT - 59T^{2} \)
61 \( 1 - 1.77iT - 61T^{2} \)
67 \( 1 - 13.5iT - 67T^{2} \)
71 \( 1 + 3.08T + 71T^{2} \)
73 \( 1 + 1.90T + 73T^{2} \)
79 \( 1 + 1.98T + 79T^{2} \)
83 \( 1 + 2.21iT - 83T^{2} \)
89 \( 1 + 6.33T + 89T^{2} \)
97 \( 1 + 14.5T + 97T^{2} \)
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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

−8.541248927049876773133603760916, −7.68802573613403919029079092881, −7.20103498844480061300659882755, −6.71627020267695337056517829264, −5.91881364052666610100755187597, −4.80101566304435703874625434725, −4.06114051408760625519687817766, −2.90689153183871802958036460083, −2.33076714037990729847190071929, −1.12678110508293559624363244650, 0.942925172228745075400159966096, 1.49352531766003419573540796955, 3.07842185215297013289046633352, 4.03600070275447352852794143401, 4.91301714559662830028145704845, 5.08420343801753754628360847032, 6.13714804938713584472024145198, 7.15890643173546327699480917706, 8.000525686117944375959500286385, 8.639091707048351204460419778327

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