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Properties

Label 2-1350-45.2-c1-0-10
Degree $2$
Conductor $1350$
Sign $0.843 + 0.537i$
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  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−4.57 + 1.22i)7-s + (−0.707 − 0.707i)8-s + (−3 − 1.73i)11-s + (4.57 + 1.22i)13-s + (−2.36 − 4.09i)14-s + (0.500 − 0.866i)16-s − 3.19i·19-s + (0.896 − 3.34i)22-s + (0.568 − 2.12i)23-s + 4.73i·26-s + (3.34 − 3.34i)28-s + (5.36 − 9.29i)29-s + (−0.0980 − 0.169i)31-s + (0.965 + 0.258i)32-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.433 + 0.249i)4-s + (−1.72 + 0.462i)7-s + (−0.249 − 0.249i)8-s + (−0.904 − 0.522i)11-s + (1.26 + 0.339i)13-s + (−0.632 − 1.09i)14-s + (0.125 − 0.216i)16-s − 0.733i·19-s + (0.191 − 0.713i)22-s + (0.118 − 0.442i)23-s + 0.928i·26-s + (0.632 − 0.632i)28-s + (0.996 − 1.72i)29-s + (−0.0176 − 0.0305i)31-s + (0.170 + 0.0457i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.843 + 0.537i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.843 + 0.537i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9264269752\)
\(L(\frac12)\) \(\approx\) \(0.9264269752\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (4.57 - 1.22i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.57 - 1.22i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 + 3.19iT - 19T^{2} \)
23 \( 1 + (-0.568 + 2.12i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-5.36 + 9.29i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.0980 + 0.169i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.79 - 5.79i)T + 37iT^{2} \)
41 \( 1 + (-1.5 + 0.866i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.120 + 0.448i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.55 + 5.79i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (5.79 + 5.79i)T + 53iT^{2} \)
59 \( 1 + (2.76 + 4.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.43 - 5.34i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.26iT - 71T^{2} \)
73 \( 1 + (3.67 - 3.67i)T - 73iT^{2} \)
79 \( 1 + (8.66 + 5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-16.6 + 4.45i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 8.66T + 89T^{2} \)
97 \( 1 + (2.56 - 0.688i)T + (84.0 - 48.5i)T^{2} \)
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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

−9.459315460263006837609030289059, −8.619504428732301928653057423069, −8.013669465926624749056722842404, −6.75803586137454990156473202241, −6.30618901148573749043173142105, −5.64381957709818648979377291974, −4.46656173934679330108677601956, −3.39618747906898522597586928233, −2.64217034944461984353757988534, −0.40388732302127486146344689973, 1.16743597125724983301604010061, 2.78205479271066931751110697021, 3.41754390298525452445190795347, 4.31883004103203764373936585611, 5.55953655931306290694286413342, 6.26417138812670356657417534383, 7.18399879260228509624148423820, 8.150068906018501172804690330050, 9.162379159013128252811350766551, 9.767304287717341223326142519825

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