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Properties

Label 2-1350-45.4-c1-0-0
Degree $2$
Conductor $1350$
Sign $-0.980 - 0.195i$
Analytic cond. $10.7798$
Root an. cond. $3.28326$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (0.389 − 0.224i)7-s + 0.999i·8-s + (−2.44 − 4.24i)11-s + (0.389 + 0.224i)13-s + (−0.224 + 0.389i)14-s + (−0.5 − 0.866i)16-s + 4.89i·17-s − 7.44·19-s + (4.24 + 2.44i)22-s + (−2.12 − 1.22i)23-s − 0.449·26-s − 0.449i·28-s + (1.22 + 2.12i)29-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.147 − 0.0849i)7-s + 0.353i·8-s + (−0.738 − 1.27i)11-s + (0.107 + 0.0623i)13-s + (−0.0600 + 0.104i)14-s + (−0.125 − 0.216i)16-s + 1.18i·17-s − 1.70·19-s + (0.904 + 0.522i)22-s + (−0.442 − 0.255i)23-s − 0.0881·26-s − 0.0849i·28-s + (0.227 + 0.393i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.980 - 0.195i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ -0.980 - 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2130683417\)
\(L(\frac12)\) \(\approx\) \(0.2130683417\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.389 + 0.224i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.44 + 4.24i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.389 - 0.224i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.89iT - 17T^{2} \)
19 \( 1 + 7.44T + 19T^{2} \)
23 \( 1 + (2.12 + 1.22i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.22 - 2.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.22 - 3.85i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 11.3iT - 37T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.20 + 1.27i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (9.43 - 5.44i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.55iT - 53T^{2} \)
59 \( 1 + (-2.72 + 4.71i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.301 - 0.174i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 + iT - 73T^{2} \)
79 \( 1 + (-8.34 - 14.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.71 + 2.72i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (7.61 - 4.39i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.04182254225589312621368954788, −8.917825423181372554618526657774, −8.339490578109850870881137555081, −7.86835728255680159585753695233, −6.56416748692617554562181564673, −6.12498154022965176259003090909, −5.11141767083487073126422444153, −3.99685387688611163999115370024, −2.80411736353517922731003428963, −1.51733003721263135997844622854, 0.10246353154306206935961105424, 1.91121521712688236787892436783, 2.64111953416455343705167776431, 4.05124396753137062726543635823, 4.85561133051351852743234338798, 5.99030135531057856854626751787, 7.03425187799792393197020344952, 7.67200150899748574084827668781, 8.453505232763313032449521991846, 9.373442301791414041863858157950

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