Properties

Label 2-3360-5.4-c1-0-52
Degree $2$
Conductor $3360$
Sign $-0.447 + 0.894i$
Analytic cond. $26.8297$
Root an. cond. $5.17974$
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 + (−2 − i)5-s i·7-s − 9-s + 4·11-s − 2i·13-s + (−1 + 2i)15-s − 6i·17-s + 6·19-s − 21-s + 2i·23-s + (3 + 4i)25-s + i·27-s + 6·29-s − 2·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.894 − 0.447i)5-s − 0.377i·7-s − 0.333·9-s + 1.20·11-s − 0.554i·13-s + (−0.258 + 0.516i)15-s − 1.45i·17-s + 1.37·19-s − 0.218·21-s + 0.417i·23-s + (0.600 + 0.800i)25-s + 0.192i·27-s + 1.11·29-s − 0.359·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(26.8297\)
Root analytic conductor: \(5.17974\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (2689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.613148800\)
\(L(\frac12)\) \(\approx\) \(1.613148800\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
5 \( 1 + (2 + i)T \)
7 \( 1 + iT \)
good11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 - 2iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−8.218989491862241874310563928510, −7.55288325098702594165502106098, −7.10805706004313871067638185457, −6.26122938090300512744682722937, −5.23231614210226916337449333408, −4.56893854666655810046859401015, −3.55966027357382780450031188523, −2.89422433923791605270181446126, −1.33082579950994232297089636410, −0.61356664369414126430990298668, 1.17505176232683934023706049905, 2.56027479478351291638804228502, 3.55108049990210595287526687804, 4.09016668267637507937135721745, 4.84014988807504099471872870284, 6.04780182219825164866892677847, 6.45812107844451681419746120773, 7.51226570486987670801716930340, 8.064299758356909863675232984616, 9.090278864648666465341224446487

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