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Properties

Label 2-1350-45.34-c1-0-6
Degree $2$
Conductor $1350$
Sign $0.958 - 0.285i$
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 + (1.73 + i)7-s − 0.999i·8-s + (−1.5 + 2.59i)11-s + (1.73 − i)13-s + (−0.999 − 1.73i)14-s + (−0.5 + 0.866i)16-s − 3i·17-s + 19-s + (2.59 − 1.5i)22-s + (5.19 − 3i)23-s − 1.99·26-s + 1.99i·28-s + (−3 + 5.19i)29-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.654 + 0.377i)7-s − 0.353i·8-s + (−0.452 + 0.783i)11-s + (0.480 − 0.277i)13-s + (−0.267 − 0.462i)14-s + (−0.125 + 0.216i)16-s − 0.727i·17-s + 0.229·19-s + (0.553 − 0.319i)22-s + (1.08 − 0.625i)23-s − 0.392·26-s + 0.377i·28-s + (−0.557 + 0.964i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.958 - 0.285i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (1099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.958 - 0.285i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.304396884\)
\(L(\frac12)\) \(\approx\) \(1.304396884\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 - i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.73 + i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + (-5.19 + 3i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.19 - 3i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.33 - 2.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 11iT - 73T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-10.3 - 6i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (-4.33 - 2.5i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.617771800612619279685443925838, −8.857271304762629900855333946097, −8.155103059003509256532380302557, −7.36136740865326923198019330648, −6.57772348216509778509032095579, −5.29883376164088731735363203912, −4.64808054925628873923713421850, −3.27270078000422020160256890644, −2.33896801028541606784872011459, −1.12354455649615385936801076877, 0.809417021435246842165111118282, 2.07486634272720704910137362105, 3.46177340284659046461767636430, 4.53322895786870081684100637872, 5.63260456470012453038019217777, 6.22495984260923507135459695052, 7.46791877186840891202829811530, 7.81374963001493664111570013529, 8.814790349975111706863269139596, 9.323900950959999098263713190530

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