Properties

Label 2-8016-1.1-c1-0-34
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 + 4.19·5-s − 1.43·7-s + 9-s − 5.55·11-s − 2.46·13-s − 4.19·15-s − 0.151·17-s − 0.370·19-s + 1.43·21-s + 5.22·23-s + 12.6·25-s − 27-s − 9.43·29-s − 6.03·31-s + 5.55·33-s − 6.01·35-s + 9.91·37-s + 2.46·39-s − 8.03·41-s + 12.3·43-s + 4.19·45-s + 5.19·47-s − 4.94·49-s + 0.151·51-s + 6.02·53-s − 23.3·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.87·5-s − 0.541·7-s + 0.333·9-s − 1.67·11-s − 0.683·13-s − 1.08·15-s − 0.0368·17-s − 0.0849·19-s + 0.312·21-s + 1.08·23-s + 2.52·25-s − 0.192·27-s − 1.75·29-s − 1.08·31-s + 0.966·33-s − 1.01·35-s + 1.62·37-s + 0.394·39-s − 1.25·41-s + 1.88·43-s + 0.625·45-s + 0.757·47-s − 0.706·49-s + 0.0212·51-s + 0.827·53-s − 3.14·55-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

\(L(1)\) \(\approx\) \(1.751571692\)
\(L(\frac12)\) \(\approx\) \(1.751571692\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.19T + 5T^{2} \)
7 \( 1 + 1.43T + 7T^{2} \)
11 \( 1 + 5.55T + 11T^{2} \)
13 \( 1 + 2.46T + 13T^{2} \)
17 \( 1 + 0.151T + 17T^{2} \)
19 \( 1 + 0.370T + 19T^{2} \)
23 \( 1 - 5.22T + 23T^{2} \)
29 \( 1 + 9.43T + 29T^{2} \)
31 \( 1 + 6.03T + 31T^{2} \)
37 \( 1 - 9.91T + 37T^{2} \)
41 \( 1 + 8.03T + 41T^{2} \)
43 \( 1 - 12.3T + 43T^{2} \)
47 \( 1 - 5.19T + 47T^{2} \)
53 \( 1 - 6.02T + 53T^{2} \)
59 \( 1 - 6.13T + 59T^{2} \)
61 \( 1 + 4.70T + 61T^{2} \)
67 \( 1 - 6.05T + 67T^{2} \)
71 \( 1 + 3.38T + 71T^{2} \)
73 \( 1 + 13.2T + 73T^{2} \)
79 \( 1 - 5.34T + 79T^{2} \)
83 \( 1 + 1.83T + 83T^{2} \)
89 \( 1 - 7.69T + 89T^{2} \)
97 \( 1 - 18.7T + 97T^{2} \)
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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

−7.45966877260441634478995547307, −7.24058034876925340233788451633, −6.17168216956088652114120082493, −5.75994230358819945101196904218, −5.24447766351789670591487747243, −4.64148103102347071241035434042, −3.29731819657770232180853934534, −2.46324807194289657055643983702, −1.95857005234615926869685949532, −0.64893896729074391839953749161, 0.64893896729074391839953749161, 1.95857005234615926869685949532, 2.46324807194289657055643983702, 3.29731819657770232180853934534, 4.64148103102347071241035434042, 5.24447766351789670591487747243, 5.75994230358819945101196904218, 6.17168216956088652114120082493, 7.24058034876925340233788451633, 7.45966877260441634478995547307

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