Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \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  + 3-s − 4.01·5-s − 3.41·7-s + 9-s − 0.942·11-s − 0.0524·13-s − 4.01·15-s + 2.46·17-s + 6.45·19-s − 3.41·21-s − 1.91·23-s + 11.1·25-s + 27-s − 1.42·29-s + 3.96·31-s − 0.942·33-s + 13.6·35-s + 6.45·37-s − 0.0524·39-s + 4.79·41-s + 3.95·43-s − 4.01·45-s − 7.81·47-s + 4.63·49-s + 2.46·51-s − 5.37·53-s + 3.78·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.79·5-s − 1.28·7-s + 0.333·9-s − 0.284·11-s − 0.0145·13-s − 1.03·15-s + 0.598·17-s + 1.48·19-s − 0.744·21-s − 0.399·23-s + 2.22·25-s + 0.192·27-s − 0.264·29-s + 0.711·31-s − 0.164·33-s + 2.31·35-s + 1.06·37-s − 0.00840·39-s + 0.748·41-s + 0.603·43-s − 0.598·45-s − 1.13·47-s + 0.662·49-s + 0.345·51-s − 0.737·53-s + 0.510·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6036\)    =    \(2^{2} \cdot 3 \cdot 503\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6036} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6036,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;503\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;503\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - T \)
503 \( 1 + T \)
good5 \( 1 + 4.01T + 5T^{2} \)
7 \( 1 + 3.41T + 7T^{2} \)
11 \( 1 + 0.942T + 11T^{2} \)
13 \( 1 + 0.0524T + 13T^{2} \)
17 \( 1 - 2.46T + 17T^{2} \)
19 \( 1 - 6.45T + 19T^{2} \)
23 \( 1 + 1.91T + 23T^{2} \)
29 \( 1 + 1.42T + 29T^{2} \)
31 \( 1 - 3.96T + 31T^{2} \)
37 \( 1 - 6.45T + 37T^{2} \)
41 \( 1 - 4.79T + 41T^{2} \)
43 \( 1 - 3.95T + 43T^{2} \)
47 \( 1 + 7.81T + 47T^{2} \)
53 \( 1 + 5.37T + 53T^{2} \)
59 \( 1 - 1.63T + 59T^{2} \)
61 \( 1 + 3.15T + 61T^{2} \)
67 \( 1 + 2.49T + 67T^{2} \)
71 \( 1 - 3.47T + 71T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 + 5.58T + 79T^{2} \)
83 \( 1 + 6.10T + 83T^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 + 4.98T + 97T^{2} \)
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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

−7.73434807587309712420192672337, −7.26454127084855206715813622022, −6.48764960112485770583874886598, −5.58525620490046462036626848572, −4.55001192517843629690277352340, −3.90844136799464090212766440168, −3.15636945108994463480548913567, −2.83733697042501530409433191032, −1.08996770885750032809396032239, 0, 1.08996770885750032809396032239, 2.83733697042501530409433191032, 3.15636945108994463480548913567, 3.90844136799464090212766440168, 4.55001192517843629690277352340, 5.58525620490046462036626848572, 6.48764960112485770583874886598, 7.26454127084855206715813622022, 7.73434807587309712420192672337

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