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Properties

Label 2-3360-105.83-c0-0-1
Degree $2$
Conductor $3360$
Sign $-0.850 - 0.525i$
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 + 2i·11-s − 1.00·15-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 + (−1.41 − 1.41i)33-s − 1.00i·35-s + (−1 + i)37-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 + 2i·11-s − 1.00·15-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 + (−1.41 − 1.41i)33-s − 1.00i·35-s + (−1 + i)37-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.850 - 0.525i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (3233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :0),\ -0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7043776608\)
\(L(\frac12)\) \(\approx\) \(0.7043776608\)
\(L(1)\) 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 \( 1 - 2iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - iT^{2} \)
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 \( 1 - T^{2} \)
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

−9.369780317474119681144765835169, −8.563295353373662238432008469777, −7.16275923523562737296268947862, −6.75953727230032611673439721124, −6.37698000436167948237482238110, −5.12502765818217918523432446890, −4.63581582410514797538010248763, −3.72857078874404254294913784706, −2.78534157540500777561613026980, −1.62325491945416098093834723918, 0.44389406066761480172771924739, 1.72102022850308997771877137893, 2.68033255828806444843775289344, 3.72948967295506741411680163173, 5.03781726724227775418070512940, 5.65541706252530411636655841555, 6.11647133080889287714858957776, 6.70818270074833301138431958767, 7.81602273521110090539129012609, 8.729686590010766303149266081473

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