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Properties

Label 2-3360-105.44-c0-0-5
Degree $2$
Conductor $3360$
Sign $0.0633 + 0.997i$
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  + (−0.258 − 0.965i)3-s + (0.866 + 0.5i)5-s + (−0.965 − 0.258i)7-s + (−0.866 + 0.499i)9-s + (0.258 − 0.965i)15-s + i·21-s + (0.965 − 1.67i)23-s + (0.499 + 0.866i)25-s + (0.707 + 0.707i)27-s − 1.73i·29-s + (−0.707 − 0.707i)35-s − i·41-s − 0.517i·43-s − 45-s + (0.707 − 1.22i)47-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)3-s + (0.866 + 0.5i)5-s + (−0.965 − 0.258i)7-s + (−0.866 + 0.499i)9-s + (0.258 − 0.965i)15-s + i·21-s + (0.965 − 1.67i)23-s + (0.499 + 0.866i)25-s + (0.707 + 0.707i)27-s − 1.73i·29-s + (−0.707 − 0.707i)35-s − i·41-s − 0.517i·43-s − 45-s + (0.707 − 1.22i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.0633 + 0.997i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (1409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :0),\ 0.0633 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.095346619\)
\(L(\frac12)\) \(\approx\) \(1.095346619\)
\(L(1)\) 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 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (0.965 + 0.258i)T \)
good11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + 1.73iT - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + iT - T^{2} \)
43 \( 1 + 0.517iT - T^{2} \)
47 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (1.67 - 0.965i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - 0.517T + T^{2} \)
89 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.673556042274562128781190492905, −7.68269020454488004985161973747, −6.91708713906391027428099070879, −6.48196837398867891318800568668, −5.86785683141724257745575513689, −5.03578906578232891754411890274, −3.78423519032791927515790176156, −2.70685446278997619806693725951, −2.17643689812507960871961271041, −0.71573768746270268383465830780, 1.31982921631893166719888410208, 2.79948632957793143150540680119, 3.38871198146869784701233093348, 4.48711472013428768134678984595, 5.25867186554097803106995730847, 5.81445821738415613435806852400, 6.50783947023065626288227380581, 7.42190452434640059928436426493, 8.618887341857127329938512244826, 9.222047414715562816193305605342

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