Properties

Label 2-8016-1.1-c1-0-111
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 + 4.21·5-s + 5.13·7-s + 9-s + 1.59·11-s + 5.05·13-s − 4.21·15-s + 0.754·17-s + 1.66·19-s − 5.13·21-s − 6.04·23-s + 12.7·25-s − 27-s − 4.24·29-s + 1.07·31-s − 1.59·33-s + 21.6·35-s + 4.94·37-s − 5.05·39-s + 3.04·41-s − 12.5·43-s + 4.21·45-s − 9.35·47-s + 19.3·49-s − 0.754·51-s − 8.52·53-s + 6.70·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.88·5-s + 1.94·7-s + 0.333·9-s + 0.479·11-s + 1.40·13-s − 1.08·15-s + 0.183·17-s + 0.382·19-s − 1.12·21-s − 1.25·23-s + 2.54·25-s − 0.192·27-s − 0.788·29-s + 0.192·31-s − 0.277·33-s + 3.65·35-s + 0.813·37-s − 0.809·39-s + 0.475·41-s − 1.91·43-s + 0.627·45-s − 1.36·47-s + 2.77·49-s − 0.105·51-s − 1.17·53-s + 0.903·55-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

\(L(1)\) \(\approx\) \(3.878645912\)
\(L(\frac12)\) \(\approx\) \(3.878645912\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.21T + 5T^{2} \)
7 \( 1 - 5.13T + 7T^{2} \)
11 \( 1 - 1.59T + 11T^{2} \)
13 \( 1 - 5.05T + 13T^{2} \)
17 \( 1 - 0.754T + 17T^{2} \)
19 \( 1 - 1.66T + 19T^{2} \)
23 \( 1 + 6.04T + 23T^{2} \)
29 \( 1 + 4.24T + 29T^{2} \)
31 \( 1 - 1.07T + 31T^{2} \)
37 \( 1 - 4.94T + 37T^{2} \)
41 \( 1 - 3.04T + 41T^{2} \)
43 \( 1 + 12.5T + 43T^{2} \)
47 \( 1 + 9.35T + 47T^{2} \)
53 \( 1 + 8.52T + 53T^{2} \)
59 \( 1 - 3.98T + 59T^{2} \)
61 \( 1 - 8.18T + 61T^{2} \)
67 \( 1 - 4.99T + 67T^{2} \)
71 \( 1 + 6.73T + 71T^{2} \)
73 \( 1 + 6.06T + 73T^{2} \)
79 \( 1 - 13.5T + 79T^{2} \)
83 \( 1 - 5.95T + 83T^{2} \)
89 \( 1 - 7.83T + 89T^{2} \)
97 \( 1 - 3.42T + 97T^{2} \)
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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

−8.106115238035057224832942788729, −6.92109814365061402078161227023, −6.27369122367421358675140156839, −5.71320141865055513425246588105, −5.21097038699393545449149816761, −4.54473218456069645374582586097, −3.61221725470657292064265806420, −2.27901904670486966073280333389, −1.57868081602622836298929187307, −1.21267317792877513237501782715, 1.21267317792877513237501782715, 1.57868081602622836298929187307, 2.27901904670486966073280333389, 3.61221725470657292064265806420, 4.54473218456069645374582586097, 5.21097038699393545449149816761, 5.71320141865055513425246588105, 6.27369122367421358675140156839, 6.92109814365061402078161227023, 8.106115238035057224832942788729

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