Properties

Degree $2$
Conductor $3264$
Sign $-0.970 + 0.242i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + 4i·7-s − 9-s + 4i·11-s − 2·13-s + (1 + 4i)17-s − 4·19-s − 4·21-s + 4i·23-s + 5·25-s i·27-s + 4i·31-s − 4·33-s − 8i·37-s − 2i·39-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.51i·7-s − 0.333·9-s + 1.20i·11-s − 0.554·13-s + (0.242 + 0.970i)17-s − 0.917·19-s − 0.872·21-s + 0.834i·23-s + 25-s − 0.192i·27-s + 0.718i·31-s − 0.696·33-s − 1.31i·37-s − 0.320i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3264\)    =    \(2^{6} \cdot 3 \cdot 17\)
Sign: $-0.970 + 0.242i$
Motivic weight: \(1\)
Character: $\chi_{3264} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3264,\ (\ :1/2),\ -0.970 + 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.114495186\)
\(L(\frac12)\) \(\approx\) \(1.114495186\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - iT \)
17 \( 1 + (-1 - 4i)T \)
good5 \( 1 - 5T^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + 8iT - 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 8iT - 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 16iT - 97T^{2} \)
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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

−9.092073775247950372142565643745, −8.504995662132083264407473647956, −7.61971616911127348160666198782, −6.75928121739465375726192172993, −5.87322303603784433446890478681, −5.23750171066785105713896496118, −4.52526555564160521048747049391, −3.57119675450040322613192029112, −2.50410773825798860514505138230, −1.82518497361076507274090842118, 0.35720249118236664948040314366, 1.17429122779692212625963207886, 2.59924538469102992238637666025, 3.37721044627627627731585389728, 4.40825594732626851832215490337, 5.07841413722666243729852725725, 6.31409620129345260214814525694, 6.67519169143584120937097545513, 7.54589818118301820357274453582, 8.093918147859020171507176024935

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