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Properties

Label 2-2160-45.34-c1-0-12
Degree $2$
Conductor $2160$
Sign $0.933 + 0.358i$
Analytic cond. $17.2476$
Root an. cond. $4.15303$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.03 + 0.917i)5-s + (−3.85 − 2.22i)7-s + (−0.724 + 1.25i)11-s + (−2.12 + 1.22i)13-s + 3.89i·17-s − 0.550·19-s + (−2.51 + 1.44i)23-s + (3.31 − 3.74i)25-s + (3 − 5.19i)29-s + (3.22 + 5.58i)31-s + (9.89 + i)35-s − 8i·37-s + (−0.5 − 0.866i)41-s + (6.45 + 3.72i)43-s + (0.389 + 0.224i)47-s + ⋯
L(s)  = 1  + (−0.911 + 0.410i)5-s + (−1.45 − 0.840i)7-s + (−0.218 + 0.378i)11-s + (−0.588 + 0.339i)13-s + 0.945i·17-s − 0.126·19-s + (−0.523 + 0.302i)23-s + (0.663 − 0.748i)25-s + (0.557 − 0.964i)29-s + (0.579 + 1.00i)31-s + (1.67 + 0.169i)35-s − 1.31i·37-s + (−0.0780 − 0.135i)41-s + (0.983 + 0.568i)43-s + (0.0567 + 0.0327i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2160\)    =    \(2^{4} \cdot 3^{3} \cdot 5\)
Sign: $0.933 + 0.358i$
Analytic conductor: \(17.2476\)
Root analytic conductor: \(4.15303\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2160} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2160,\ (\ :1/2),\ 0.933 + 0.358i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8018493288\)
\(L(\frac12)\) \(\approx\) \(0.8018493288\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (2.03 - 0.917i)T \)
good7 \( 1 + (3.85 + 2.22i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.724 - 1.25i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.12 - 1.22i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.89iT - 17T^{2} \)
19 \( 1 + 0.550T + 19T^{2} \)
23 \( 1 + (2.51 - 1.44i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.22 - 5.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.45 - 3.72i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.389 - 0.224i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.44iT - 53T^{2} \)
59 \( 1 + (5.62 + 9.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.224 + 0.389i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.18 + 4.72i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.44T + 71T^{2} \)
73 \( 1 + 4.79iT - 73T^{2} \)
79 \( 1 + (-3.67 + 6.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.46 - 2i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 + (-11.2 - 6.5i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.133611474235237919748132500997, −8.028885432446324064814743624746, −7.50165442647459623309324815528, −6.65316494888786218298874528701, −6.22901151281846699901545721312, −4.82519770663759980571272118326, −3.94210055158892111975332027808, −3.40685569695766528692202166300, −2.29108391647043236371949649477, −0.48056110538368925821822310963, 0.64990898907417036929084595833, 2.61078723162431865432533730832, 3.14870411431948936390523607510, 4.22513327981055042458809882659, 5.15007977891585833585725028746, 5.97554714872557344281459902092, 6.80112326574468894078572035567, 7.59078934153124004292363728388, 8.430005332178260896092962297473, 9.087243326331460437670789482681

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