Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 167 $
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 + 3.18·5-s + 1.39·7-s + 9-s + 2.54·11-s + 5.18·13-s − 3.18·15-s − 2.90·17-s − 6.55·19-s − 1.39·21-s − 2.68·23-s + 5.12·25-s − 27-s − 3.50·29-s − 6.85·31-s − 2.54·33-s + 4.43·35-s + 4.23·37-s − 5.18·39-s − 1.88·41-s + 6.49·43-s + 3.18·45-s + 7.80·47-s − 5.05·49-s + 2.90·51-s + 10.5·53-s + 8.10·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.42·5-s + 0.526·7-s + 0.333·9-s + 0.767·11-s + 1.43·13-s − 0.821·15-s − 0.704·17-s − 1.50·19-s − 0.304·21-s − 0.559·23-s + 1.02·25-s − 0.192·27-s − 0.650·29-s − 1.23·31-s − 0.443·33-s + 0.749·35-s + 0.696·37-s − 0.829·39-s − 0.295·41-s + 0.990·43-s + 0.474·45-s + 1.13·47-s − 0.722·49-s + 0.406·51-s + 1.44·53-s + 1.09·55-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

\( d \)  =  \(2\)
\( N \)  =  \(8016\)    =    \(2^{4} \cdot 3 \cdot 167\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8016} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.652229708$
$L(\frac12)$  $\approx$  $2.652229708$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;167\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;167\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 + T \)
167 \( 1 - T \)
good5 \( 1 - 3.18T + 5T^{2} \)
7 \( 1 - 1.39T + 7T^{2} \)
11 \( 1 - 2.54T + 11T^{2} \)
13 \( 1 - 5.18T + 13T^{2} \)
17 \( 1 + 2.90T + 17T^{2} \)
19 \( 1 + 6.55T + 19T^{2} \)
23 \( 1 + 2.68T + 23T^{2} \)
29 \( 1 + 3.50T + 29T^{2} \)
31 \( 1 + 6.85T + 31T^{2} \)
37 \( 1 - 4.23T + 37T^{2} \)
41 \( 1 + 1.88T + 41T^{2} \)
43 \( 1 - 6.49T + 43T^{2} \)
47 \( 1 - 7.80T + 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 + 0.200T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 - 2.92T + 71T^{2} \)
73 \( 1 - 14.6T + 73T^{2} \)
79 \( 1 + 1.95T + 79T^{2} \)
83 \( 1 - 3.16T + 83T^{2} \)
89 \( 1 - 8.01T + 89T^{2} \)
97 \( 1 - 2.21T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.87857704774685537534208298974, −6.74605122035881477732357952922, −6.45660932155325547772468249966, −5.75191966703741569361392894169, −5.31560463212375852186512354486, −4.19433286703833022709617506473, −3.80687699677767572902560433045, −2.24457811171838443975036012364, −1.86359497112076174620157233746, −0.861476369774250741556415320596, 0.861476369774250741556415320596, 1.86359497112076174620157233746, 2.24457811171838443975036012364, 3.80687699677767572902560433045, 4.19433286703833022709617506473, 5.31560463212375852186512354486, 5.75191966703741569361392894169, 6.45660932155325547772468249966, 6.74605122035881477732357952922, 7.87857704774685537534208298974

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