Properties

Degree 2
Conductor $ 2^{4} \cdot 503 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.823·3-s − 2.12·5-s − 4.15·7-s − 2.32·9-s − 0.347·11-s + 3.83·13-s + 1.74·15-s − 6.72·17-s + 6.76·19-s + 3.42·21-s − 0.816·23-s − 0.494·25-s + 4.38·27-s + 1.71·29-s − 2.32·31-s + 0.285·33-s + 8.81·35-s + 5.48·37-s − 3.16·39-s − 1.70·41-s + 5.51·43-s + 4.92·45-s + 6.07·47-s + 10.2·49-s + 5.54·51-s + 3.81·53-s + 0.737·55-s + ⋯
L(s)  = 1  − 0.475·3-s − 0.949·5-s − 1.57·7-s − 0.773·9-s − 0.104·11-s + 1.06·13-s + 0.451·15-s − 1.63·17-s + 1.55·19-s + 0.746·21-s − 0.170·23-s − 0.0988·25-s + 0.843·27-s + 0.318·29-s − 0.418·31-s + 0.0497·33-s + 1.49·35-s + 0.902·37-s − 0.506·39-s − 0.266·41-s + 0.840·43-s + 0.734·45-s + 0.885·47-s + 1.46·49-s + 0.775·51-s + 0.524·53-s + 0.0993·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8048\)    =    \(2^{4} \cdot 503\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8048} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8048,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;503\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;503\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
503 \( 1 + T \)
good3 \( 1 + 0.823T + 3T^{2} \)
5 \( 1 + 2.12T + 5T^{2} \)
7 \( 1 + 4.15T + 7T^{2} \)
11 \( 1 + 0.347T + 11T^{2} \)
13 \( 1 - 3.83T + 13T^{2} \)
17 \( 1 + 6.72T + 17T^{2} \)
19 \( 1 - 6.76T + 19T^{2} \)
23 \( 1 + 0.816T + 23T^{2} \)
29 \( 1 - 1.71T + 29T^{2} \)
31 \( 1 + 2.32T + 31T^{2} \)
37 \( 1 - 5.48T + 37T^{2} \)
41 \( 1 + 1.70T + 41T^{2} \)
43 \( 1 - 5.51T + 43T^{2} \)
47 \( 1 - 6.07T + 47T^{2} \)
53 \( 1 - 3.81T + 53T^{2} \)
59 \( 1 - 8.10T + 59T^{2} \)
61 \( 1 + 15.2T + 61T^{2} \)
67 \( 1 - 7.22T + 67T^{2} \)
71 \( 1 - 8.48T + 71T^{2} \)
73 \( 1 + 4.46T + 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 - 1.00T + 83T^{2} \)
89 \( 1 + 8.30T + 89T^{2} \)
97 \( 1 - 15.0T + 97T^{2} \)
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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

−7.35519761794097744424899880402, −6.76620950860045877798818031272, −6.05781126158527216271558008820, −5.64576590655301005515091464901, −4.55922776483689132792349542555, −3.80535493985572714622088264184, −3.22460940654024536403958030398, −2.46308905071911455893621312274, −0.850138043771136430203071057853, 0, 0.850138043771136430203071057853, 2.46308905071911455893621312274, 3.22460940654024536403958030398, 3.80535493985572714622088264184, 4.55922776483689132792349542555, 5.64576590655301005515091464901, 6.05781126158527216271558008820, 6.76620950860045877798818031272, 7.35519761794097744424899880402

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