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Properties

Label 2-1620-9.4-c3-0-33
Degree $2$
Conductor $1620$
Sign $0.939 + 0.342i$
Analytic cond. $95.5830$
Root an. cond. $9.77666$
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  + (2.5 − 4.33i)5-s + (14 + 24.2i)7-s + (12 + 20.7i)11-s + (35 − 60.6i)13-s + 102·17-s + 20·19-s + (36 − 62.3i)23-s + (−12.5 − 21.6i)25-s + (−153 − 265. i)29-s + (68 − 117. i)31-s + 140·35-s − 214·37-s + (75 − 129. i)41-s + (146 + 252. i)43-s + (36 + 62.3i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (0.755 + 1.30i)7-s + (0.328 + 0.569i)11-s + (0.746 − 1.29i)13-s + 1.45·17-s + 0.241·19-s + (0.326 − 0.565i)23-s + (−0.100 − 0.173i)25-s + (−0.979 − 1.69i)29-s + (0.393 − 0.682i)31-s + 0.676·35-s − 0.950·37-s + (0.285 − 0.494i)41-s + (0.517 + 0.896i)43-s + (0.111 + 0.193i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(95.5830\)
Root analytic conductor: \(9.77666\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :3/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.925288977\)
\(L(\frac12)\) \(\approx\) \(2.925288977\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-2.5 + 4.33i)T \)
good7 \( 1 + (-14 - 24.2i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-12 - 20.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-35 + 60.6i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 102T + 4.91e3T^{2} \)
19 \( 1 - 20T + 6.85e3T^{2} \)
23 \( 1 + (-36 + 62.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (153 + 265. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-68 + 117. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 214T + 5.06e4T^{2} \)
41 \( 1 + (-75 + 129. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-146 - 252. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-36 - 62.3i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 414T + 1.48e5T^{2} \)
59 \( 1 + (-372 + 644. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-209 - 361. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (94 - 162. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 480T + 3.57e5T^{2} \)
73 \( 1 - 434T + 3.89e5T^{2} \)
79 \( 1 + (676 + 1.17e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-306 - 530. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 30T + 7.04e5T^{2} \)
97 \( 1 + (-143 - 247. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−9.004950288388443492818748560512, −8.038271361946161501970497748082, −7.80420787189111440276171276538, −6.32354982596190584996862678230, −5.59943108236760566824742908738, −5.11547745629316050356351775688, −3.93901412510655582070102294841, −2.81209350492732998515621274386, −1.84197804878582479252471801470, −0.75947991292065423044074001193, 1.05097088206094220398406702252, 1.64826094086859136519938954158, 3.33980812878794745722594568203, 3.84371858473859105265148134215, 4.95052222643614238358329786420, 5.80833882506336601245316338744, 6.91549750502524689844509254828, 7.28341902959714912689409168425, 8.273532226706615999655988689460, 9.058430558841057384794197740581

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