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Properties

Label 2-1350-5.4-c1-0-23
Degree $2$
Conductor $1350$
Sign $-0.447 - 0.894i$
Analytic cond. $10.7798$
Root an. cond. $3.28326$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − 4-s − 2i·7-s + i·8-s − 3·11-s − i·13-s − 2·14-s + 16-s + 3i·17-s − 8·19-s + 3i·22-s + 3i·23-s − 26-s + 2i·28-s + 9·29-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 0.755i·7-s + 0.353i·8-s − 0.904·11-s − 0.277i·13-s − 0.534·14-s + 0.250·16-s + 0.727i·17-s − 1.83·19-s + 0.639i·22-s + 0.625i·23-s − 0.196·26-s + 0.377i·28-s + 1.67·29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 - 3iT - 23T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 12T + 41T^{2} \)
43 \( 1 + 7iT - 43T^{2} \)
47 \( 1 - 3iT - 47T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 - T + 79T^{2} \)
83 \( 1 - 18iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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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

−9.069985098066405319787624372977, −8.352372513268844398044555900659, −7.59252296923371189433624452265, −6.58995008955294092268364781725, −5.58951178643922843514037220930, −4.58228423763220223600684148784, −3.81073648314959636717886844386, −2.73758707498326604068581247919, −1.58799533461463586566106427761, 0, 2.06987808139532852517291433380, 3.14726929746407118576421448585, 4.54809523429238690758711146532, 5.10777863684652661076409099255, 6.18442251921636355924927443594, 6.73641744432083828405596959665, 7.80781508822472564125868734871, 8.525820122150584578788359468372, 9.057006075543861612881565421457

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