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Properties

Label 2-3360-105.62-c0-0-0
Degree $2$
Conductor $3360$
Sign $-0.525 - 0.850i$
Analytic cond. $1.67685$
Root an. cond. $1.29493$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)3-s + (−0.707 − 0.707i)5-s + (−0.707 + 0.707i)7-s + 1.00i·9-s − 1.00i·15-s + (1.41 + 1.41i)17-s − 1.41·19-s − 1.00·21-s + (−1 − i)23-s + 1.00i·25-s + (−0.707 + 0.707i)27-s + 1.41i·31-s + 1.00·35-s + (1 + i)37-s − 1.41·41-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s + (−0.707 − 0.707i)5-s + (−0.707 + 0.707i)7-s + 1.00i·9-s − 1.00i·15-s + (1.41 + 1.41i)17-s − 1.41·19-s − 1.00·21-s + (−1 − i)23-s + 1.00i·25-s + (−0.707 + 0.707i)27-s + 1.41i·31-s + 1.00·35-s + (1 + i)37-s − 1.41·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.525 - 0.850i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (1217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :0),\ -0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9738653440\)
\(L(\frac12)\) \(\approx\) \(0.9738653440\)
\(L(1)\) 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 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good11 \( 1 - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
19 \( 1 + 1.41T + T^{2} \)
23 \( 1 + (1 + i)T + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 1.41iT - T^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 + 1.41T + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 2iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−8.788480309901355348589718034493, −8.382692510544872679309752573399, −8.035267091361205260211748305195, −6.80896370279099142041000967867, −5.95129184767025811922896185261, −5.14125631528036360454889034407, −4.25121629032857723869808032716, −3.65805592110341863055821231420, −2.81701798163101577554049097297, −1.66989948474539997603693503365, 0.51267878725346465465903439096, 2.06096572333113468675196477700, 3.04569477009691955604367886586, 3.63189351152470652678099765184, 4.37651429734154422247156624427, 5.83101266171108791195686761391, 6.47422340273375297653353459868, 7.31007780879409605649140581558, 7.62887699043896466917023439513, 8.284588015155818561483837380060

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