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Properties

Label 2-3360-105.44-c0-0-3
Degree $2$
Conductor $3360$
Sign $0.553 + 0.832i$
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 + (−0.866 − 0.5i)5-s + (0.965 + 0.258i)7-s + 1.00i·9-s + (0.258 + 0.965i)15-s + (−0.500 − 0.866i)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 + (0.500 − 0.866i)45-s + (0.707 − 1.22i)47-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s + (−0.866 − 0.5i)5-s + (0.965 + 0.258i)7-s + 1.00i·9-s + (0.258 + 0.965i)15-s + (−0.500 − 0.866i)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 + (0.500 − 0.866i)45-s + (0.707 − 1.22i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.553 + 0.832i$
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: \((2,\ 3360,\ (\ :0),\ 0.553 + 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9301073578\)
\(L(\frac12)\) \(\approx\) \(0.9301073578\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T \)
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

−8.372531246827003233452564964769, −8.091480608347679610619989437777, −7.09871303243200929292689868194, −6.66367161891495647736908821457, −5.46398902049437037129658451774, −4.95290126382460039326441329380, −4.31434718663493027268162139050, −3.02664281931911211285422845476, −1.85402857033447995527481432060, −0.827950920659224598987851089116, 1.00817362582173333841271347532, 2.57072517571473936731330734536, 3.77685863384287038622048380194, 4.14608791307704292850145181854, 5.11883969788006943112800091790, 5.71333117411461480449743591469, 6.76751668395362576814971885870, 7.41818380278400556190591771594, 8.077211456871562482145235934627, 8.954861630252526144722599692284

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