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Properties

Label 2-3360-840.173-c0-0-0
Degree $2$
Conductor $3360$
Sign $-0.104 - 0.994i$
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  + (−0.965 − 0.258i)3-s + (−0.258 + 0.965i)5-s + (−0.5 − 0.866i)7-s + (0.866 + 0.499i)9-s + (0.965 + 1.67i)11-s + (0.499 − 0.866i)15-s + (0.258 + 0.965i)21-s + (−0.866 − 0.499i)25-s + (−0.707 − 0.707i)27-s − 0.517i·29-s + (−0.866 + 0.5i)31-s + (−0.500 − 1.86i)33-s + (0.965 − 0.258i)35-s + (−0.707 + 0.707i)45-s + (−0.499 + 0.866i)49-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)3-s + (−0.258 + 0.965i)5-s + (−0.5 − 0.866i)7-s + (0.866 + 0.499i)9-s + (0.965 + 1.67i)11-s + (0.499 − 0.866i)15-s + (0.258 + 0.965i)21-s + (−0.866 − 0.499i)25-s + (−0.707 − 0.707i)27-s − 0.517i·29-s + (−0.866 + 0.5i)31-s + (−0.500 − 1.86i)33-s + (0.965 − 0.258i)35-s + (−0.707 + 0.707i)45-s + (−0.499 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.104 - 0.994i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :0),\ -0.104 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6602504540\)
\(L(\frac12)\) \(\approx\) \(0.6602504540\)
\(L(1)\) 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 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 + (0.258 - 0.965i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good11 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + 0.517iT - T^{2} \)
31 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.866 - 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \)
59 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
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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

−9.296312516196570359871108774886, −7.81994460570321739804432744143, −7.32717060755596924872915268003, −6.71600128620784907856362751675, −6.34765360154881181367262935265, −5.21383023432447458549181974115, −4.23451979723249153391911358628, −3.80187004979224333549825441693, −2.43180662113229724116396174826, −1.32858538081087236172080148476, 0.48699632001152393262441606572, 1.69103313593907169033149994456, 3.31271821361239535831865483192, 3.91162753231950467754782082180, 4.96978375088771027149637296549, 5.60783230375929014836544933542, 6.15258022080364591420649895914, 6.86597545840592127342688515460, 8.030291315339130425683878578859, 8.760383839020327665153878521981

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