Properties

Label 2-3360-105.74-c0-0-7
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: $\chi_{3360} (3329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :0),\ -0.0633 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.709292343\)
\(L(\frac12)\) \(\approx\) \(1.709292343\)
\(L(1)\) 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

−8.485160497189785135512760073430, −8.056868946771393468630288798186, −7.01985255672697251656384912516, −6.51133993663749231667371270871, −5.56807749772854513018227917887, −4.96614691673927239117209768428, −3.95285663889900818693210704024, −2.65517240350598773487149643906, −1.90862499772350490124523117414, −1.04057673472036721774141244882, 1.76319210846271758349500120481, 2.50329197888624356224929205153, 3.54365403883430324923092362188, 4.35537609784439611353900669960, 5.31852031421457396777609971016, 5.71650879980373000560719936272, 6.62979739172028667040735112761, 7.86753431482088266811542554214, 8.108236009971729565425277505568, 9.329365364357899190561372826612

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