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Properties

Label 2-1350-45.34-c1-0-5
Degree $2$
Conductor $1350$
Sign $-0.496 - 0.867i$
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 + (2.92 + 1.68i)7-s + 0.999i·8-s + (−2.18 + 3.78i)11-s + (−5.84 + 3.37i)13-s + (1.68 + 2.92i)14-s + (−0.5 + 0.866i)16-s − 1.62i·17-s + 2.37·19-s + (−3.78 + 2.18i)22-s + (−1.18 + 0.686i)23-s − 6.74·26-s + 3.37i·28-s + (−0.686 + 1.18i)29-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (1.10 + 0.637i)7-s + 0.353i·8-s + (−0.659 + 1.14i)11-s + (−1.61 + 0.935i)13-s + (0.450 + 0.780i)14-s + (−0.125 + 0.216i)16-s − 0.394i·17-s + 0.544·19-s + (−0.807 + 0.466i)22-s + (−0.247 + 0.143i)23-s − 1.32·26-s + 0.637i·28-s + (−0.127 + 0.220i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.496 - 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.496 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.496 - 0.867i$
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: \((2,\ 1350,\ (\ :1/2),\ -0.496 - 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.122955382\)
\(L(\frac12)\) \(\approx\) \(2.122955382\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.92 - 1.68i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.18 - 3.78i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.84 - 3.37i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.62iT - 17T^{2} \)
19 \( 1 - 2.37T + 19T^{2} \)
23 \( 1 + (1.18 - 0.686i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.686 - 1.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.37 + 4.10i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.87 - 2.81i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.38 - 3.68i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.4iT - 53T^{2} \)
59 \( 1 + (2.18 + 3.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.05 + 7.02i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.06 + 3.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 3.11iT - 73T^{2} \)
79 \( 1 + (-1 + 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.38 - 3.68i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 16.1T + 89T^{2} \)
97 \( 1 + (7.25 + 4.18i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.690865337687145659157619759552, −9.156096669263746772318998372385, −7.85924165029792975395812899887, −7.54992477247677946793900045115, −6.66016299107569586667492998404, −5.38461511216715716777333357701, −4.94145591615495339122304826272, −4.20260730699176640138231499980, −2.61362542045687639282899471636, −1.95682745269629563300845312724, 0.67972239734047404614014648393, 2.15519019057184065534929319212, 3.16390162769281882420786764955, 4.20196542038691277560214381718, 5.22841184356057072923986586088, 5.56124366054076807877193323211, 6.95791939104962573878860582530, 7.74077384140480356847525934844, 8.325330668829385913011978253773, 9.521684961289127956683829937655

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