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Properties

Label 2-3360-12.11-c1-0-56
Degree $2$
Conductor $3360$
Sign $0.999 - 0.00162i$
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  + (−1.22 + 1.22i)3-s + i·5-s + i·7-s + (−0.00973 − 2.99i)9-s + 0.528·11-s + 5.03·13-s + (−1.22 − 1.22i)15-s − 3.10i·17-s + 2.74i·19-s + (−1.22 − 1.22i)21-s − 0.764·23-s − 25-s + (3.69 + 3.65i)27-s − 5.05i·29-s − 8.46i·31-s + ⋯
L(s)  = 1  + (−0.705 + 0.708i)3-s + 0.447i·5-s + 0.377i·7-s + (−0.00324 − 0.999i)9-s + 0.159·11-s + 1.39·13-s + (−0.316 − 0.315i)15-s − 0.753i·17-s + 0.629i·19-s + (−0.267 − 0.266i)21-s − 0.159·23-s − 0.200·25-s + (0.710 + 0.703i)27-s − 0.939i·29-s − 1.51i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00162i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.999 - 0.00162i$
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: \((2,\ 3360,\ (\ :1/2),\ 0.999 - 0.00162i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.379703798\)
\(L(\frac12)\) \(\approx\) \(1.379703798\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 - iT \)
7 \( 1 - iT \)
good11 \( 1 - 0.528T + 11T^{2} \)
13 \( 1 - 5.03T + 13T^{2} \)
17 \( 1 + 3.10iT - 17T^{2} \)
19 \( 1 - 2.74iT - 19T^{2} \)
23 \( 1 + 0.764T + 23T^{2} \)
29 \( 1 + 5.05iT - 29T^{2} \)
31 \( 1 + 8.46iT - 31T^{2} \)
37 \( 1 - 2.30T + 37T^{2} \)
41 \( 1 + 2.11iT - 41T^{2} \)
43 \( 1 + 2.69iT - 43T^{2} \)
47 \( 1 + 1.32T + 47T^{2} \)
53 \( 1 + 14.0iT - 53T^{2} \)
59 \( 1 - 4.90T + 59T^{2} \)
61 \( 1 + 7.31T + 61T^{2} \)
67 \( 1 + 9.70iT - 67T^{2} \)
71 \( 1 + 7.16T + 71T^{2} \)
73 \( 1 - 9.83T + 73T^{2} \)
79 \( 1 - 3.17iT - 79T^{2} \)
83 \( 1 + 1.76T + 83T^{2} \)
89 \( 1 + 6.81iT - 89T^{2} \)
97 \( 1 - 3.31T + 97T^{2} \)
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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

−8.692576170014885427880042248987, −7.947664450360579196560555951471, −6.95917038840359307956601840573, −6.08502312068590343889716494573, −5.82967414881899495738318305723, −4.77870113974428266332953562240, −3.91609847020413020031125117654, −3.29351965297964644973210895598, −2.03902829714452995774449281028, −0.57964623720249431235626871639, 0.999593058681730867023771620588, 1.61909367475249354238673038153, 3.01702827462444781458168612896, 4.07346561244534950672797988572, 4.86987285008270131821043506091, 5.73022424343787777292253246783, 6.37103636667304658223736173960, 7.01843325365179487188993800905, 7.86015958704026825601342268177, 8.559028911285261269685258610807

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