Properties

Degree 2
Conductor $ 2^{4} \cdot 251 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.844·3-s − 2.47·5-s + 0.972·7-s − 2.28·9-s − 3.50·11-s + 2.87·13-s + 2.08·15-s − 2.27·17-s + 6.78·19-s − 0.821·21-s − 3.58·23-s + 1.10·25-s + 4.46·27-s − 7.60·29-s + 0.279·31-s + 2.96·33-s − 2.40·35-s − 9.37·37-s − 2.43·39-s − 10.4·41-s + 5.70·43-s + 5.64·45-s + 8.12·47-s − 6.05·49-s + 1.91·51-s + 1.01·53-s + 8.66·55-s + ⋯
L(s)  = 1  − 0.487·3-s − 1.10·5-s + 0.367·7-s − 0.762·9-s − 1.05·11-s + 0.798·13-s + 0.538·15-s − 0.550·17-s + 1.55·19-s − 0.179·21-s − 0.746·23-s + 0.220·25-s + 0.859·27-s − 1.41·29-s + 0.0502·31-s + 0.516·33-s − 0.405·35-s − 1.54·37-s − 0.389·39-s − 1.63·41-s + 0.869·43-s + 0.841·45-s + 1.18·47-s − 0.864·49-s + 0.268·51-s + 0.139·53-s + 1.16·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4016\)    =    \(2^{4} \cdot 251\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4016} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6979239047$
$L(\frac12)$  $\approx$  $0.6979239047$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;251\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;251\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
251 \( 1 + T \)
good3 \( 1 + 0.844T + 3T^{2} \)
5 \( 1 + 2.47T + 5T^{2} \)
7 \( 1 - 0.972T + 7T^{2} \)
11 \( 1 + 3.50T + 11T^{2} \)
13 \( 1 - 2.87T + 13T^{2} \)
17 \( 1 + 2.27T + 17T^{2} \)
19 \( 1 - 6.78T + 19T^{2} \)
23 \( 1 + 3.58T + 23T^{2} \)
29 \( 1 + 7.60T + 29T^{2} \)
31 \( 1 - 0.279T + 31T^{2} \)
37 \( 1 + 9.37T + 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 - 5.70T + 43T^{2} \)
47 \( 1 - 8.12T + 47T^{2} \)
53 \( 1 - 1.01T + 53T^{2} \)
59 \( 1 + 12.3T + 59T^{2} \)
61 \( 1 - 9.95T + 61T^{2} \)
67 \( 1 - 4.50T + 67T^{2} \)
71 \( 1 + 9.82T + 71T^{2} \)
73 \( 1 - 6.37T + 73T^{2} \)
79 \( 1 + 1.13T + 79T^{2} \)
83 \( 1 - 1.53T + 83T^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
97 \( 1 + 0.254T + 97T^{2} \)
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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

−8.364505739749856292781866864550, −7.70398176588759232527318391811, −7.18393360834196490463609784824, −6.09436195928041855663939883124, −5.42429836030108780293639255764, −4.82319647199216930539129532985, −3.74154557439109419882114565654, −3.17743047874291837502981265778, −1.92817789147770722494401165372, −0.47402549824066669448991644431, 0.47402549824066669448991644431, 1.92817789147770722494401165372, 3.17743047874291837502981265778, 3.74154557439109419882114565654, 4.82319647199216930539129532985, 5.42429836030108780293639255764, 6.09436195928041855663939883124, 7.18393360834196490463609784824, 7.70398176588759232527318391811, 8.364505739749856292781866864550

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