Properties

Degree $2$
Conductor $8016$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 0.588·5-s + 1.23·7-s + 9-s + 5.01·11-s + 4.86·13-s − 0.588·15-s − 6.70·17-s + 1.72·19-s − 1.23·21-s − 3.90·23-s − 4.65·25-s − 27-s + 8.98·29-s + 5.36·31-s − 5.01·33-s + 0.726·35-s + 1.07·37-s − 4.86·39-s + 6.22·41-s + 9.74·43-s + 0.588·45-s − 3.85·47-s − 5.47·49-s + 6.70·51-s − 7.57·53-s + 2.95·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.263·5-s + 0.466·7-s + 0.333·9-s + 1.51·11-s + 1.34·13-s − 0.151·15-s − 1.62·17-s + 0.395·19-s − 0.269·21-s − 0.814·23-s − 0.930·25-s − 0.192·27-s + 1.66·29-s + 0.964·31-s − 0.873·33-s + 0.122·35-s + 0.175·37-s − 0.778·39-s + 0.972·41-s + 1.48·43-s + 0.0877·45-s − 0.562·47-s − 0.782·49-s + 0.939·51-s − 1.04·53-s + 0.398·55-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: $1$
Motivic weight: \(1\)
Character: $\chi_{8016} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8016,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.280585969\)
\(L(\frac12)\) \(\approx\) \(2.280585969\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.588T + 5T^{2} \)
7 \( 1 - 1.23T + 7T^{2} \)
11 \( 1 - 5.01T + 11T^{2} \)
13 \( 1 - 4.86T + 13T^{2} \)
17 \( 1 + 6.70T + 17T^{2} \)
19 \( 1 - 1.72T + 19T^{2} \)
23 \( 1 + 3.90T + 23T^{2} \)
29 \( 1 - 8.98T + 29T^{2} \)
31 \( 1 - 5.36T + 31T^{2} \)
37 \( 1 - 1.07T + 37T^{2} \)
41 \( 1 - 6.22T + 41T^{2} \)
43 \( 1 - 9.74T + 43T^{2} \)
47 \( 1 + 3.85T + 47T^{2} \)
53 \( 1 + 7.57T + 53T^{2} \)
59 \( 1 - 8.99T + 59T^{2} \)
61 \( 1 + 4.59T + 61T^{2} \)
67 \( 1 - 15.2T + 67T^{2} \)
71 \( 1 - 6.37T + 71T^{2} \)
73 \( 1 - 8.12T + 73T^{2} \)
79 \( 1 - 6.10T + 79T^{2} \)
83 \( 1 + 3.46T + 83T^{2} \)
89 \( 1 - 5.15T + 89T^{2} \)
97 \( 1 + 13.6T + 97T^{2} \)
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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

−8.029102635186803786093553726398, −6.78962221079480448195972506986, −6.43440866748595494184992653135, −5.98782062698145011641645401375, −4.99827639402586904953165433080, −4.19636412705898407268617012963, −3.84830805366332631722219124519, −2.53431060417193783515299842258, −1.59381477912860457584154167299, −0.836329489761701077791381687715, 0.836329489761701077791381687715, 1.59381477912860457584154167299, 2.53431060417193783515299842258, 3.84830805366332631722219124519, 4.19636412705898407268617012963, 4.99827639402586904953165433080, 5.98782062698145011641645401375, 6.43440866748595494184992653135, 6.78962221079480448195972506986, 8.029102635186803786093553726398

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