Properties

Label 2-4400-5.4-c1-0-34
Degree $2$
Conductor $4400$
Sign $-0.447 - 0.894i$
Analytic cond. $35.1341$
Root an. cond. $5.92740$
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  + 3i·3-s − 2i·7-s − 6·9-s + 11-s + 6i·17-s + 4·19-s + 6·21-s i·23-s − 9i·27-s + 8·29-s + 7·31-s + 3i·33-s + i·37-s + 4·41-s − 6i·43-s + ⋯
L(s)  = 1  + 1.73i·3-s − 0.755i·7-s − 2·9-s + 0.301·11-s + 1.45i·17-s + 0.917·19-s + 1.30·21-s − 0.208i·23-s − 1.73i·27-s + 1.48·29-s + 1.25·31-s + 0.522i·33-s + 0.164i·37-s + 0.624·41-s − 0.914i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4400\)    =    \(2^{4} \cdot 5^{2} \cdot 11\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(35.1341\)
Root analytic conductor: \(5.92740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4400} (4049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4400,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.901789022\)
\(L(\frac12)\) \(\approx\) \(1.901789022\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 - T \)
good3 \( 1 - 3iT - 3T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + iT - 23T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 - iT - 37T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 + 5iT - 67T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 - 16iT - 73T^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 - 2iT - 83T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 - 7iT - 97T^{2} \)
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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

−8.591158913313704180525562879618, −8.210438531955501446103592766434, −7.09981035968807620092504832390, −6.26457842295921585138265576609, −5.47895167395053194509541917472, −4.69262992226800341002375474446, −4.05569850635683108654609145824, −3.54331587082993759977051842019, −2.57937755004782498892208386134, −1.00102072246229447522872074416, 0.67020504408855193567243288286, 1.48746709163632682711682285924, 2.69487002432059801616202550248, 2.91708226982043547549471154008, 4.53001543068568994012316134246, 5.39381535128777363073752519237, 6.12672976890198425998295788291, 6.70388946021883691305078858621, 7.41209920928680676984447504326, 7.951431383324858364901210758269

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