Properties

Label 2-4056-1.1-c1-0-49
Degree $2$
Conductor $4056$
Sign $1$
Analytic cond. $32.3873$
Root an. cond. $5.69098$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2.72·5-s + 3.31·7-s + 9-s + 0.656·11-s + 2.72·15-s − 1.16·17-s + 6.77·19-s + 3.31·21-s + 6.90·23-s + 2.41·25-s + 27-s − 5.82·29-s − 0.969·31-s + 0.656·33-s + 9.02·35-s − 9.93·37-s + 10.5·41-s − 5.07·43-s + 2.72·45-s − 8.24·47-s + 3.99·49-s − 1.16·51-s + 0.841·53-s + 1.78·55-s + 6.77·57-s + 0.128·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.21·5-s + 1.25·7-s + 0.333·9-s + 0.197·11-s + 0.702·15-s − 0.283·17-s + 1.55·19-s + 0.723·21-s + 1.43·23-s + 0.482·25-s + 0.192·27-s − 1.08·29-s − 0.174·31-s + 0.114·33-s + 1.52·35-s − 1.63·37-s + 1.65·41-s − 0.773·43-s + 0.405·45-s − 1.20·47-s + 0.570·49-s − 0.163·51-s + 0.115·53-s + 0.241·55-s + 0.896·57-s + 0.0166·59-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(3.854063766\)
\(L(\frac12)\) \(\approx\) \(3.854063766\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.72T + 5T^{2} \)
7 \( 1 - 3.31T + 7T^{2} \)
11 \( 1 - 0.656T + 11T^{2} \)
17 \( 1 + 1.16T + 17T^{2} \)
19 \( 1 - 6.77T + 19T^{2} \)
23 \( 1 - 6.90T + 23T^{2} \)
29 \( 1 + 5.82T + 29T^{2} \)
31 \( 1 + 0.969T + 31T^{2} \)
37 \( 1 + 9.93T + 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 + 5.07T + 43T^{2} \)
47 \( 1 + 8.24T + 47T^{2} \)
53 \( 1 - 0.841T + 53T^{2} \)
59 \( 1 - 0.128T + 59T^{2} \)
61 \( 1 - 11.7T + 61T^{2} \)
67 \( 1 + 15.9T + 67T^{2} \)
71 \( 1 - 5.32T + 71T^{2} \)
73 \( 1 + 3.75T + 73T^{2} \)
79 \( 1 - 2.17T + 79T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 + 4.09T + 89T^{2} \)
97 \( 1 - 16.5T + 97T^{2} \)
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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

−8.625482649295338227739327969820, −7.61253307417027621497342350423, −7.18409268476166208588916602616, −6.17655896972092836337074616176, −5.25684264035894549344661493865, −4.95109796859476632747362621707, −3.75031715504071794551370802059, −2.82368236504562179420383336656, −1.86312191754174594161417444205, −1.26540526978308159887043557136, 1.26540526978308159887043557136, 1.86312191754174594161417444205, 2.82368236504562179420383336656, 3.75031715504071794551370802059, 4.95109796859476632747362621707, 5.25684264035894549344661493865, 6.17655896972092836337074616176, 7.18409268476166208588916602616, 7.61253307417027621497342350423, 8.625482649295338227739327969820

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