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Properties

Label 2-3360-840.389-c0-0-0
Degree $2$
Conductor $3360$
Sign $-0.997 - 0.0633i$
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  + (−0.866 − 0.5i)3-s + i·5-s + (−0.5 + 0.866i)7-s + (0.499 + 0.866i)9-s + (−0.866 + 1.5i)11-s + (0.5 − 0.866i)15-s + (0.866 − 0.499i)21-s − 25-s − 0.999i·27-s − 1.73·29-s + (0.5 − 0.866i)31-s + (1.5 − 0.866i)33-s + (−0.866 − 0.5i)35-s + (−0.866 + 0.499i)45-s + (−0.499 − 0.866i)49-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + i·5-s + (−0.5 + 0.866i)7-s + (0.499 + 0.866i)9-s + (−0.866 + 1.5i)11-s + (0.5 − 0.866i)15-s + (0.866 − 0.499i)21-s − 25-s − 0.999i·27-s − 1.73·29-s + (0.5 − 0.866i)31-s + (1.5 − 0.866i)33-s + (−0.866 − 0.5i)35-s + (−0.866 + 0.499i)45-s + (−0.499 − 0.866i)49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.997 - 0.0633i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (1649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :0),\ -0.997 - 0.0633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3444219365\)
\(L(\frac12)\) \(\approx\) \(0.3444219365\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 - iT \)
7 \( 1 + (0.5 - 0.866i)T \)
good11 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + 1.73T + T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - iT - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - 1.73iT - T^{2} \)
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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

−9.461932143595083949931809500532, −8.117544928299998900582394344207, −7.51996803469776318838975272327, −6.89444322985583510334351638371, −6.22034492476518892717897659256, −5.51202500529077856338661037780, −4.79466723302134564493009361319, −3.66137186410304829870240862333, −2.44973243318153198766099807534, −1.98132035817400889090460418254, 0.23366001741253895896962949046, 1.30573304650371896594606997984, 3.07041191798465419482441483614, 3.87305855505506616878973324106, 4.61284434646938418228667398628, 5.53229203373696301998289634732, 5.86961653013272374268652792076, 6.87375195616329808010894907652, 7.70517662559185341051148732955, 8.522609627768367972869877201521

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