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Properties

Label 2-3360-40.29-c1-0-46
Degree $2$
Conductor $3360$
Sign $0.316 + 0.948i$
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  + 3-s + (−2 − i)5-s − i·7-s + 9-s + 4i·11-s + 2·13-s + (−2 − i)15-s − 2i·17-s − 8i·19-s − i·21-s + 4i·23-s + (3 + 4i)25-s + 27-s − 2i·29-s + 4·31-s + ⋯
L(s)  = 1  + 0.577·3-s + (−0.894 − 0.447i)5-s − 0.377i·7-s + 0.333·9-s + 1.20i·11-s + 0.554·13-s + (−0.516 − 0.258i)15-s − 0.485i·17-s − 1.83i·19-s − 0.218i·21-s + 0.834i·23-s + (0.600 + 0.800i)25-s + 0.192·27-s − 0.371i·29-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.316 + 0.948i$
Analytic conductor: \(26.8297\)
Root analytic conductor: \(5.17974\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :1/2),\ 0.316 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.737996728\)
\(L(\frac12)\) \(\approx\) \(1.737996728\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 + (2 + i)T \)
7 \( 1 + iT \)
good11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 8iT - 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 + 14T + 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 + 10T + 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 - 8T + 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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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

−8.426336635636305127068186430565, −7.68130236284686718342587077275, −7.19264089521570009848293430474, −6.48689932135419456135463669843, −5.06674516919571137340226387986, −4.61824952685902356533292830462, −3.78978205021579893172733102290, −2.95233662161984075967218719510, −1.81906811780195090324792399110, −0.56381971390923017164732230783, 1.10170041188640126478914951421, 2.41801035713232971077379923375, 3.41455003614794314970890911207, 3.76595964921642079534572064373, 4.81489010576000963676604277736, 6.05003892972168458098315915763, 6.34892621300295132023060141861, 7.55835377641132013487815061062, 8.098669702489921424113953329183, 8.567819942799249714602010885128

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