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Properties

Label 2-3328-8.5-c1-0-41
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  + 2i·5-s + 2·7-s + 3·9-s − 2i·11-s + i·13-s + 6·17-s + 6i·19-s − 8·23-s + 25-s − 2i·29-s + 10·31-s + 4i·35-s − 6i·37-s + 6·41-s + 4i·43-s + ⋯
L(s)  = 1  + 0.894i·5-s + 0.755·7-s + 9-s − 0.603i·11-s + 0.277i·13-s + 1.45·17-s + 1.37i·19-s − 1.66·23-s + 0.200·25-s − 0.371i·29-s + 1.79·31-s + 0.676i·35-s − 0.986i·37-s + 0.937·41-s + 0.609i·43-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

\(L(1)\) \(\approx\) \(2.391387640\)
\(L(\frac12)\) \(\approx\) \(2.391387640\)
\(L(\frac{3}{2})\) 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 \( 1 - 3T^{2} \)
5 \( 1 - 2iT - 5T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 10iT - 59T^{2} \)
61 \( 1 - 2iT - 61T^{2} \)
67 \( 1 + 10iT - 67T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
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

−8.511218615322201476125324886202, −7.77674216922836956532674511703, −7.51739694025035837376805256081, −6.27766473415904451839885481166, −5.97648036585378566945259649141, −4.80514747541410603419948792692, −3.99040548654153492098683079604, −3.24384143830868426317446865170, −2.10261292904118788852418234774, −1.14906112456943752191491702303, 0.894354999349218799491935799062, 1.67257766671392312969540112048, 2.85609335182729288188206440351, 4.13920696068713736577114544821, 4.68164800387422122711270661767, 5.27643298662796075001281096075, 6.27567747924984960660861854308, 7.20964060717829026776974784389, 7.85052329033276788798317621047, 8.435339999531146319969007556001

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