Properties

Label 2-8016-1.1-c1-0-106
Degree $2$
Conductor $8016$
Sign $1$
Analytic cond. $64.0080$
Root an. cond. $8.00050$
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 + 3.69·5-s + 2.24·7-s + 9-s + 5.56·13-s + 3.69·15-s − 1.01·17-s + 2.65·19-s + 2.24·21-s − 2.65·23-s + 8.62·25-s + 27-s − 1.38·29-s − 4.89·31-s + 8.27·35-s − 1.97·37-s + 5.56·39-s + 4.28·41-s − 0.551·43-s + 3.69·45-s + 5.62·47-s − 1.97·49-s − 1.01·51-s + 1.03·53-s + 2.65·57-s + 2.42·59-s + 5.38·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.65·5-s + 0.847·7-s + 0.333·9-s + 1.54·13-s + 0.953·15-s − 0.246·17-s + 0.608·19-s + 0.489·21-s − 0.552·23-s + 1.72·25-s + 0.192·27-s − 0.256·29-s − 0.878·31-s + 1.39·35-s − 0.325·37-s + 0.891·39-s + 0.669·41-s − 0.0840·43-s + 0.550·45-s + 0.820·47-s − 0.281·49-s − 0.142·51-s + 0.142·53-s + 0.351·57-s + 0.316·59-s + 0.689·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8016\)    =    \(2^{4} \cdot 3 \cdot 167\)
Sign: $1$
Analytic conductor: \(64.0080\)
Root analytic conductor: \(8.00050\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8016} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8016,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.678106528\)
\(L(\frac12)\) \(\approx\) \(4.678106528\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
167 \( 1 - T \)
good5 \( 1 - 3.69T + 5T^{2} \)
7 \( 1 - 2.24T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 5.56T + 13T^{2} \)
17 \( 1 + 1.01T + 17T^{2} \)
19 \( 1 - 2.65T + 19T^{2} \)
23 \( 1 + 2.65T + 23T^{2} \)
29 \( 1 + 1.38T + 29T^{2} \)
31 \( 1 + 4.89T + 31T^{2} \)
37 \( 1 + 1.97T + 37T^{2} \)
41 \( 1 - 4.28T + 41T^{2} \)
43 \( 1 + 0.551T + 43T^{2} \)
47 \( 1 - 5.62T + 47T^{2} \)
53 \( 1 - 1.03T + 53T^{2} \)
59 \( 1 - 2.42T + 59T^{2} \)
61 \( 1 - 5.38T + 61T^{2} \)
67 \( 1 + 14.8T + 67T^{2} \)
71 \( 1 - 8.68T + 71T^{2} \)
73 \( 1 + 1.75T + 73T^{2} \)
79 \( 1 + 3.74T + 79T^{2} \)
83 \( 1 - 5.43T + 83T^{2} \)
89 \( 1 + 6.27T + 89T^{2} \)
97 \( 1 + 11.0T + 97T^{2} \)
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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

−7.935415353416980516557592632042, −7.14968084155472886833228888867, −6.33201177498050209060347066930, −5.74951495554562112382130368846, −5.20849405937760450944233503296, −4.25249171471864487990168816978, −3.45651291624323972820686779589, −2.48335168892101473527837465214, −1.76242458560598234595187591527, −1.18316245138188585173662989672, 1.18316245138188585173662989672, 1.76242458560598234595187591527, 2.48335168892101473527837465214, 3.45651291624323972820686779589, 4.25249171471864487990168816978, 5.20849405937760450944233503296, 5.74951495554562112382130368846, 6.33201177498050209060347066930, 7.14968084155472886833228888867, 7.935415353416980516557592632042

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