Properties

Label 2-3360-12.11-c1-0-95
Degree $2$
Conductor $3360$
Sign $-0.291 - 0.956i$
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  + (0.814 − 1.52i)3-s i·5-s i·7-s + (−1.67 − 2.49i)9-s − 1.02·11-s − 4.05·13-s + (−1.52 − 0.814i)15-s + 3.17i·17-s − 2.43i·19-s + (−1.52 − 0.814i)21-s − 6.89·23-s − 25-s + (−5.16 + 0.527i)27-s + 4.16i·29-s + 2.76i·31-s + ⋯
L(s)  = 1  + (0.470 − 0.882i)3-s − 0.447i·5-s − 0.377i·7-s + (−0.557 − 0.830i)9-s − 0.310·11-s − 1.12·13-s + (−0.394 − 0.210i)15-s + 0.770i·17-s − 0.558i·19-s + (−0.333 − 0.177i)21-s − 1.43·23-s − 0.200·25-s + (−0.994 + 0.101i)27-s + 0.772i·29-s + 0.496i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.291 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.291 - 0.956i$
Analytic conductor: \(26.8297\)
Root analytic conductor: \(5.17974\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (2591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :1/2),\ -0.291 - 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.01267571384\)
\(L(\frac12)\) \(\approx\) \(0.01267571384\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.814 + 1.52i)T \)
5 \( 1 + iT \)
7 \( 1 + iT \)
good11 \( 1 + 1.02T + 11T^{2} \)
13 \( 1 + 4.05T + 13T^{2} \)
17 \( 1 - 3.17iT - 17T^{2} \)
19 \( 1 + 2.43iT - 19T^{2} \)
23 \( 1 + 6.89T + 23T^{2} \)
29 \( 1 - 4.16iT - 29T^{2} \)
31 \( 1 - 2.76iT - 31T^{2} \)
37 \( 1 - 0.949T + 37T^{2} \)
41 \( 1 - 5.27iT - 41T^{2} \)
43 \( 1 - 9.82iT - 43T^{2} \)
47 \( 1 - 5.55T + 47T^{2} \)
53 \( 1 + 2.52iT - 53T^{2} \)
59 \( 1 - 13.4T + 59T^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 + 12.3iT - 67T^{2} \)
71 \( 1 - 0.483T + 71T^{2} \)
73 \( 1 + 2.55T + 73T^{2} \)
79 \( 1 + 4.87iT - 79T^{2} \)
83 \( 1 + 8.24T + 83T^{2} \)
89 \( 1 - 2.15iT - 89T^{2} \)
97 \( 1 - 1.82T + 97T^{2} \)
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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

−7.973185251279028735632107191715, −7.52319953431477594538661763510, −6.68076640663458723844717738741, −5.98313712983281535976686143732, −5.02279626662380436664021344610, −4.18215773729915089852185688709, −3.15721208442758697427801585567, −2.25520065007181839453527358740, −1.31866096119868334520457821564, −0.00337713108168654629613581082, 2.19777989364037772323512549240, 2.63637831657561305925345753146, 3.74814725340512940114568083188, 4.40570240068699935692148840780, 5.39573239904093630796756861838, 5.87936094077458751205394043648, 7.11820717386327282293040675578, 7.70602561121470061641254735837, 8.440101939936154309895498573604, 9.220295125186960351016325379366

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