Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 13 \cdot 31 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 1.33·3-s + 4-s − 5-s − 1.33·6-s + 1.60·7-s + 8-s − 1.22·9-s − 10-s + 3.78·11-s − 1.33·12-s + 13-s + 1.60·14-s + 1.33·15-s + 16-s − 7.50·17-s − 1.22·18-s + 1.98·19-s − 20-s − 2.14·21-s + 3.78·22-s − 7.33·23-s − 1.33·24-s + 25-s + 26-s + 5.62·27-s + 1.60·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.769·3-s + 0.5·4-s − 0.447·5-s − 0.544·6-s + 0.607·7-s + 0.353·8-s − 0.408·9-s − 0.316·10-s + 1.14·11-s − 0.384·12-s + 0.277·13-s + 0.429·14-s + 0.344·15-s + 0.250·16-s − 1.82·17-s − 0.288·18-s + 0.455·19-s − 0.223·20-s − 0.467·21-s + 0.806·22-s − 1.53·23-s − 0.272·24-s + 0.200·25-s + 0.196·26-s + 1.08·27-s + 0.303·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4030\)    =    \(2 \cdot 5 \cdot 13 \cdot 31\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4030} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4030,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;13,\;31\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;13,\;31\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
5 \( 1 + T \)
13 \( 1 - T \)
31 \( 1 + T \)
good3 \( 1 + 1.33T + 3T^{2} \)
7 \( 1 - 1.60T + 7T^{2} \)
11 \( 1 - 3.78T + 11T^{2} \)
17 \( 1 + 7.50T + 17T^{2} \)
19 \( 1 - 1.98T + 19T^{2} \)
23 \( 1 + 7.33T + 23T^{2} \)
29 \( 1 + 9.25T + 29T^{2} \)
37 \( 1 + 2.62T + 37T^{2} \)
41 \( 1 + 6.29T + 41T^{2} \)
43 \( 1 - 12.8T + 43T^{2} \)
47 \( 1 - 0.235T + 47T^{2} \)
53 \( 1 - 5.87T + 53T^{2} \)
59 \( 1 - 0.614T + 59T^{2} \)
61 \( 1 - 6.39T + 61T^{2} \)
67 \( 1 + 0.858T + 67T^{2} \)
71 \( 1 + 2.99T + 71T^{2} \)
73 \( 1 + 0.697T + 73T^{2} \)
79 \( 1 + 9.61T + 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 + 6.48T + 89T^{2} \)
97 \( 1 - 0.423T + 97T^{2} \)
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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

−7.999651280016826414565105530525, −7.09971949864646189844603325614, −6.48320623784207862424729961668, −5.79234112080214066926734029566, −5.12823526264123995728848708536, −4.15744713456309670116596082155, −3.83445539348733219093480712173, −2.47755763776379686511947491664, −1.50322302499238233980405843474, 0, 1.50322302499238233980405843474, 2.47755763776379686511947491664, 3.83445539348733219093480712173, 4.15744713456309670116596082155, 5.12823526264123995728848708536, 5.79234112080214066926734029566, 6.48320623784207862424729961668, 7.09971949864646189844603325614, 7.999651280016826414565105530525

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