Properties

Label 2-4332-1.1-c1-0-54
Degree $2$
Conductor $4332$
Sign $-1$
Analytic cond. $34.5911$
Root an. cond. $5.88142$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2.53·5-s − 3.22·7-s + 9-s + 3.10·11-s − 6.06·13-s + 2.53·15-s − 5.94·17-s − 3.22·21-s − 6.71·23-s + 1.41·25-s + 27-s + 7.12·29-s + 7.63·31-s + 3.10·33-s − 8.17·35-s + 2.10·37-s − 6.06·39-s − 7.04·41-s − 9.36·43-s + 2.53·45-s − 4.65·47-s + 3.41·49-s − 5.94·51-s − 6.35·53-s + 7.86·55-s − 0.268·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.13·5-s − 1.21·7-s + 0.333·9-s + 0.936·11-s − 1.68·13-s + 0.653·15-s − 1.44·17-s − 0.704·21-s − 1.40·23-s + 0.282·25-s + 0.192·27-s + 1.32·29-s + 1.37·31-s + 0.540·33-s − 1.38·35-s + 0.346·37-s − 0.971·39-s − 1.09·41-s − 1.42·43-s + 0.377·45-s − 0.678·47-s + 0.487·49-s − 0.832·51-s − 0.872·53-s + 1.06·55-s − 0.0349·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4332\)    =    \(2^{2} \cdot 3 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(34.5911\)
Root analytic conductor: \(5.88142\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4332} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4332,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
19 \( 1 \)
good5 \( 1 - 2.53T + 5T^{2} \)
7 \( 1 + 3.22T + 7T^{2} \)
11 \( 1 - 3.10T + 11T^{2} \)
13 \( 1 + 6.06T + 13T^{2} \)
17 \( 1 + 5.94T + 17T^{2} \)
23 \( 1 + 6.71T + 23T^{2} \)
29 \( 1 - 7.12T + 29T^{2} \)
31 \( 1 - 7.63T + 31T^{2} \)
37 \( 1 - 2.10T + 37T^{2} \)
41 \( 1 + 7.04T + 41T^{2} \)
43 \( 1 + 9.36T + 43T^{2} \)
47 \( 1 + 4.65T + 47T^{2} \)
53 \( 1 + 6.35T + 53T^{2} \)
59 \( 1 + 0.268T + 59T^{2} \)
61 \( 1 - 1.58T + 61T^{2} \)
67 \( 1 + 1.30T + 67T^{2} \)
71 \( 1 + 8.27T + 71T^{2} \)
73 \( 1 - 0.921T + 73T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 - 0.0864T + 83T^{2} \)
89 \( 1 + 7.07T + 89T^{2} \)
97 \( 1 - 3.32T + 97T^{2} \)
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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

−8.167919246004900937174280761351, −7.04704844059734516184952481446, −6.52094437352877995551163102861, −6.11626791364696536639153149592, −4.90141789711959435266160939196, −4.27381946715023235967370966879, −3.14327371152579108624660100075, −2.46979851169930612728100428583, −1.69232083052172181852257790814, 0, 1.69232083052172181852257790814, 2.46979851169930612728100428583, 3.14327371152579108624660100075, 4.27381946715023235967370966879, 4.90141789711959435266160939196, 6.11626791364696536639153149592, 6.52094437352877995551163102861, 7.04704844059734516184952481446, 8.167919246004900937174280761351

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