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 − 2.54·5-s − 0.525·7-s + 9-s − 5.59·11-s + 1.57·13-s − 2.54·15-s + 4.20·17-s − 3.78·19-s − 0.525·21-s + 2.45·23-s + 1.47·25-s + 27-s − 4.24·29-s + 5.76·31-s − 5.59·33-s + 1.33·35-s − 2.67·37-s + 1.57·39-s + 10.7·41-s − 8.36·43-s − 2.54·45-s − 4.77·47-s − 6.72·49-s + 4.20·51-s − 2.34·53-s + 14.2·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.13·5-s − 0.198·7-s + 0.333·9-s − 1.68·11-s + 0.437·13-s − 0.656·15-s + 1.02·17-s − 0.869·19-s − 0.114·21-s + 0.511·23-s + 0.294·25-s + 0.192·27-s − 0.787·29-s + 1.03·31-s − 0.973·33-s + 0.225·35-s − 0.439·37-s + 0.252·39-s + 1.68·41-s − 1.27·43-s − 0.379·45-s − 0.697·47-s − 0.960·49-s + 0.589·51-s − 0.322·53-s + 1.91·55-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  =  \((2,\ 8016,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.302237262\)
\(L(\frac12)\)  \(\approx\)  \(1.302237262\)
\(L(\frac{3}{2})\)   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
167 \( 1 + T \)
good5 \( 1 + 2.54T + 5T^{2} \)
7 \( 1 + 0.525T + 7T^{2} \)
11 \( 1 + 5.59T + 11T^{2} \)
13 \( 1 - 1.57T + 13T^{2} \)
17 \( 1 - 4.20T + 17T^{2} \)
19 \( 1 + 3.78T + 19T^{2} \)
23 \( 1 - 2.45T + 23T^{2} \)
29 \( 1 + 4.24T + 29T^{2} \)
31 \( 1 - 5.76T + 31T^{2} \)
37 \( 1 + 2.67T + 37T^{2} \)
41 \( 1 - 10.7T + 41T^{2} \)
43 \( 1 + 8.36T + 43T^{2} \)
47 \( 1 + 4.77T + 47T^{2} \)
53 \( 1 + 2.34T + 53T^{2} \)
59 \( 1 - 5.25T + 59T^{2} \)
61 \( 1 + 5.02T + 61T^{2} \)
67 \( 1 + 8.91T + 67T^{2} \)
71 \( 1 - 5.10T + 71T^{2} \)
73 \( 1 + 2.16T + 73T^{2} \)
79 \( 1 - 2.52T + 79T^{2} \)
83 \( 1 - 4.01T + 83T^{2} \)
89 \( 1 - 16.2T + 89T^{2} \)
97 \( 1 + 16.4T + 97T^{2} \)
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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

−7.977476341858645065112061466363, −7.41233647358070559119754097774, −6.57476085809938451736267980230, −5.70613200287024004362306585942, −4.90934420841920302913566478226, −4.22695995003818990183572963076, −3.35271858476553575864245098324, −2.92849046126144165695046199996, −1.87479976127938756280110250011, −0.53048879360008684140098530121, 0.53048879360008684140098530121, 1.87479976127938756280110250011, 2.92849046126144165695046199996, 3.35271858476553575864245098324, 4.22695995003818990183572963076, 4.90934420841920302913566478226, 5.70613200287024004362306585942, 6.57476085809938451736267980230, 7.41233647358070559119754097774, 7.977476341858645065112061466363

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