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Properties

Label 2-3328-8.5-c1-0-49
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  + i·3-s + i·5-s + 3·7-s + 2·9-s − 2i·11-s − i·13-s − 15-s − 3·17-s + 2i·19-s + 3i·21-s + 4·23-s + 4·25-s + 5i·27-s − 2i·29-s − 4·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.447i·5-s + 1.13·7-s + 0.666·9-s − 0.603i·11-s − 0.277i·13-s − 0.258·15-s − 0.727·17-s + 0.458i·19-s + 0.654i·21-s + 0.834·23-s + 0.800·25-s + 0.962i·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

\(L(1)\) \(\approx\) \(2.400789823\)
\(L(\frac12)\) \(\approx\) \(2.400789823\)
\(L(\frac{3}{2})\) 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 \( 1 - iT - 3T^{2} \)
5 \( 1 - iT - 5T^{2} \)
7 \( 1 - 3T + 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 5iT - 37T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 + 7iT - 43T^{2} \)
47 \( 1 - 9T + 47T^{2} \)
53 \( 1 - 4iT - 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 - 4iT - 61T^{2} \)
67 \( 1 + 10iT - 67T^{2} \)
71 \( 1 + 15T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−8.880157672028371211683869038403, −7.905114463359413762411237583990, −7.35400998102095319157005823962, −6.49476624921280522772804785712, −5.56761019982169990166111603987, −4.80604899610561460037535656113, −4.15457340430832945190843238100, −3.24796225218268772921656850650, −2.19254625144669468143236401663, −1.04747734778876854886060885548, 0.944510207715478198284809024647, 1.76602891595524199945264321345, 2.63266241948440037720055765942, 4.17543957128242856597877921997, 4.60549786053395597259857643766, 5.38607867962064949987032330061, 6.42416732087092013598387706853, 7.32767881993696853970978007488, 7.50211611712680101994387131035, 8.628387802360297444039085334453

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