Properties

Label 2-96e2-1.1-c1-0-136
Degree $2$
Conductor $9216$
Sign $-1$
Analytic cond. $73.5901$
Root an. cond. $8.57846$
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  + 1.41·5-s + 1.41·13-s − 2·17-s − 2.99·25-s − 4.24·29-s − 7.07·37-s + 10·41-s − 7·49-s − 12.7·53-s + 1.41·61-s + 2.00·65-s − 6·73-s − 2.82·85-s − 16·89-s − 8·97-s + 12.7·101-s + 9.89·109-s − 16·113-s + ⋯
L(s)  = 1  + 0.632·5-s + 0.392·13-s − 0.485·17-s − 0.599·25-s − 0.787·29-s − 1.16·37-s + 1.56·41-s − 49-s − 1.74·53-s + 0.181·61-s + 0.248·65-s − 0.702·73-s − 0.306·85-s − 1.69·89-s − 0.812·97-s + 1.26·101-s + 0.948·109-s − 1.50·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9216\)    =    \(2^{10} \cdot 3^{2}\)
Sign: $-1$
Analytic conductor: \(73.5901\)
Root analytic conductor: \(8.57846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{9216} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9216,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good5 \( 1 - 1.41T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 1.41T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 4.24T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 7.07T + 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 12.7T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 1.41T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 16T + 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−7.37685703496192901044300065441, −6.60911702221311962985694789449, −6.00572283757213578165633461735, −5.42489545218306605610735904734, −4.59917277350848073226637324373, −3.84743609344081809515585761392, −3.01445867919703423135729861720, −2.10163795549860412406988230960, −1.38807573819246094102559431375, 0, 1.38807573819246094102559431375, 2.10163795549860412406988230960, 3.01445867919703423135729861720, 3.84743609344081809515585761392, 4.59917277350848073226637324373, 5.42489545218306605610735904734, 6.00572283757213578165633461735, 6.60911702221311962985694789449, 7.37685703496192901044300065441

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