↑

Properties

Label 2-3360-105.83-c0-0-2
Degree $2$
Conductor $3360$
Sign $0.525 - 0.850i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)3-s + (0.707 − 0.707i)5-s + (0.707 + 0.707i)7-s − 1.00i·9-s + 1.00i·15-s + (−1.41 + 1.41i)17-s + 1.41·19-s − 1.00·21-s + (−1 + i)23-s − 1.00i·25-s + (0.707 + 0.707i)27-s + 1.41i·31-s + 1.00·35-s + (1 − i)37-s + 1.41·41-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s + (0.707 − 0.707i)5-s + (0.707 + 0.707i)7-s − 1.00i·9-s + 1.00i·15-s + (−1.41 + 1.41i)17-s + 1.41·19-s − 1.00·21-s + (−1 + i)23-s − 1.00i·25-s + (0.707 + 0.707i)27-s + 1.41i·31-s + 1.00·35-s + (1 − i)37-s + 1.41·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (3233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :0),\ 0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.184420400\)
\(L(\frac12)\) \(\approx\) \(1.184420400\)
\(L(1)\) 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 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good11 \( 1 - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
19 \( 1 - 1.41T + T^{2} \)
23 \( 1 + (1 - i)T - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 1.41iT - T^{2} \)
37 \( 1 + (-1 + i)T - iT^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + 2iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 + iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.114165196931640697342330676187, −8.378159477520016652842088822602, −7.50064139810412005641180398644, −6.24667371624717261620288880249, −5.87637375798843160316295720174, −5.12958166396942435341787939649, −4.51275969385245733385021408067, −3.63272617502131183661179476308, −2.26421038654408395243353952154, −1.30872187858092348319206247817, 0.862356645613774447445871601511, 2.09920709404672417074003357108, 2.75705335414895335845842959015, 4.25281533292078090539591123819, 4.91512787662359043555441658643, 5.82841111855943396181191059710, 6.43509908595508887185457534471, 7.27642380648245057903603499349, 7.56219344637655923753388254691, 8.551133033319785223672851504829

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