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Properties

Label 2-1350-5.4-c3-0-11
Degree $2$
Conductor $1350$
Sign $-0.447 + 0.894i$
Analytic cond. $79.6525$
Root an. cond. $8.92482$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 4·4-s + 30.5i·7-s − 8i·8-s − 13.5·11-s + 28.0i·13-s − 61.0·14-s + 16·16-s + 55.5i·17-s − 27.4·19-s − 27.0i·22-s + 139. i·23-s − 56.1·26-s − 122. i·28-s − 178.·29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 1.64i·7-s − 0.353i·8-s − 0.371·11-s + 0.598i·13-s − 1.16·14-s + 0.250·16-s + 0.792i·17-s − 0.331·19-s − 0.262i·22-s + 1.26i·23-s − 0.423·26-s − 0.824i·28-s − 1.14·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(79.6525\)
Root analytic conductor: \(8.92482\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :3/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9217211120\)
\(L(\frac12)\) \(\approx\) \(0.9217211120\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 30.5iT - 343T^{2} \)
11 \( 1 + 13.5T + 1.33e3T^{2} \)
13 \( 1 - 28.0iT - 2.19e3T^{2} \)
17 \( 1 - 55.5iT - 4.91e3T^{2} \)
19 \( 1 + 27.4T + 6.85e3T^{2} \)
23 \( 1 - 139. iT - 1.21e4T^{2} \)
29 \( 1 + 178.T + 2.43e4T^{2} \)
31 \( 1 - 297.T + 2.97e4T^{2} \)
37 \( 1 - 159. iT - 5.06e4T^{2} \)
41 \( 1 + 140.T + 6.89e4T^{2} \)
43 \( 1 - 5.68iT - 7.95e4T^{2} \)
47 \( 1 + 301. iT - 1.03e5T^{2} \)
53 \( 1 - 122. iT - 1.48e5T^{2} \)
59 \( 1 - 864.T + 2.05e5T^{2} \)
61 \( 1 + 47.6T + 2.26e5T^{2} \)
67 \( 1 - 402. iT - 3.00e5T^{2} \)
71 \( 1 + 927.T + 3.57e5T^{2} \)
73 \( 1 - 1.01e3iT - 3.89e5T^{2} \)
79 \( 1 + 812.T + 4.93e5T^{2} \)
83 \( 1 + 1.38e3iT - 5.71e5T^{2} \)
89 \( 1 - 1.42e3T + 7.04e5T^{2} \)
97 \( 1 + 124. iT - 9.12e5T^{2} \)
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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

−9.593087980736432797022109302825, −8.736227657100627449835677603711, −8.344642126328999653619825071293, −7.33771666044484874695343011086, −6.36119839360652819847928809857, −5.71483869658418629379363684367, −5.02172169871310044849866523393, −3.89709782159346525394716737351, −2.70728954364549824275177257075, −1.63974010500570077866732593658, 0.24208326345877002729402555958, 0.994609741045932223332476787107, 2.39680267672207565907966622920, 3.40594477052340722495809466654, 4.30678583398344722015085098398, 4.99318277837989414115082468556, 6.24015564952804046349784647741, 7.20800391115494968750584070963, 7.87598070593997435060765470612, 8.740146945959753461618771420554

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