Properties

Label 2-4056-13.12-c1-0-16
Degree $2$
Conductor $4056$
Sign $-0.554 - 0.832i$
Analytic cond. $32.3873$
Root an. cond. $5.69098$
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  + 3-s + 3i·5-s + 9-s + 3i·15-s + 17-s − 4·23-s − 4·25-s + 27-s + 3·29-s + 8i·31-s + 5i·37-s + 3i·41-s − 4·43-s + 3i·45-s + 8i·47-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34i·5-s + 0.333·9-s + 0.774i·15-s + 0.242·17-s − 0.834·23-s − 0.800·25-s + 0.192·27-s + 0.557·29-s + 1.43i·31-s + 0.821i·37-s + 0.468i·41-s − 0.609·43-s + 0.447i·45-s + 1.16i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4056\)    =    \(2^{3} \cdot 3 \cdot 13^{2}\)
Sign: $-0.554 - 0.832i$
Analytic conductor: \(32.3873\)
Root analytic conductor: \(5.69098\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4056} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4056,\ (\ :1/2),\ -0.554 - 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.879936958\)
\(L(\frac12)\) \(\approx\) \(1.879936958\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 \)
good5 \( 1 - 3iT - 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 - T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 - 8iT - 31T^{2} \)
37 \( 1 - 5iT - 37T^{2} \)
41 \( 1 - 3iT - 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + 13T + 53T^{2} \)
59 \( 1 + 12iT - 59T^{2} \)
61 \( 1 - 15T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 8iT - 71T^{2} \)
73 \( 1 + 3iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 10iT - 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−8.425129584193236544597750577012, −8.093429542561579956402647341956, −7.05666276638728578100557202365, −6.73872052276482494096933791082, −5.88622136568464825008302966386, −4.87866693708687105734821872127, −3.91059215706329356940367123656, −3.12603792166247840551936944068, −2.56538218718704606359029221767, −1.43426421156518844878516825165, 0.48705822446828856356673527654, 1.61168947646378694768543394246, 2.50964986626511634023770713047, 3.72239281218278114928859809148, 4.30795016481760430238987107373, 5.16105916834161185473654892066, 5.82707782060081975069638774171, 6.78290486579060523317340953452, 7.75162725156854576490049156413, 8.189184634703033602973399759141

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