Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 13 \cdot 31 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 1.54·3-s + 4-s − 5-s − 1.54·6-s − 2.44·7-s + 8-s − 0.600·9-s − 10-s − 5.54·11-s − 1.54·12-s − 13-s − 2.44·14-s + 1.54·15-s + 16-s + 2.34·17-s − 0.600·18-s + 4.05·19-s − 20-s + 3.79·21-s − 5.54·22-s − 9.12·23-s − 1.54·24-s + 25-s − 26-s + 5.57·27-s − 2.44·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.894·3-s + 0.5·4-s − 0.447·5-s − 0.632·6-s − 0.925·7-s + 0.353·8-s − 0.200·9-s − 0.316·10-s − 1.67·11-s − 0.447·12-s − 0.277·13-s − 0.654·14-s + 0.399·15-s + 0.250·16-s + 0.568·17-s − 0.141·18-s + 0.929·19-s − 0.223·20-s + 0.827·21-s − 1.18·22-s − 1.90·23-s − 0.316·24-s + 0.200·25-s − 0.196·26-s + 1.07·27-s − 0.462·28-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  =  0
Selberg data  =  $(2,\ 4030,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8476308832$
$L(\frac12)$  $\approx$  $0.8476308832$
$L(\frac{3}{2})$   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 \( 1 + 1.54T + 3T^{2} \)
7 \( 1 + 2.44T + 7T^{2} \)
11 \( 1 + 5.54T + 11T^{2} \)
17 \( 1 - 2.34T + 17T^{2} \)
19 \( 1 - 4.05T + 19T^{2} \)
23 \( 1 + 9.12T + 23T^{2} \)
29 \( 1 + 6.24T + 29T^{2} \)
37 \( 1 + 5.94T + 37T^{2} \)
41 \( 1 - 7.55T + 41T^{2} \)
43 \( 1 - 1.44T + 43T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 + 4.83T + 53T^{2} \)
59 \( 1 - 3.28T + 59T^{2} \)
61 \( 1 + 0.864T + 61T^{2} \)
67 \( 1 + 1.53T + 67T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 + 3.86T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 - 15.0T + 83T^{2} \)
89 \( 1 + 17.7T + 89T^{2} \)
97 \( 1 + 15.2T + 97T^{2} \)
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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

−8.091245920318994813920081512769, −7.63854946580985942742931688557, −6.86740912767840265823276104921, −5.85216368947051440602968904927, −5.62844947051291626473842549552, −4.87209727973329232221361063141, −3.83100770600986348333664653838, −3.10269842363745649311820457600, −2.23018497563588620232285605018, −0.46173468622084347996312533209, 0.46173468622084347996312533209, 2.23018497563588620232285605018, 3.10269842363745649311820457600, 3.83100770600986348333664653838, 4.87209727973329232221361063141, 5.62844947051291626473842549552, 5.85216368947051440602968904927, 6.86740912767840265823276104921, 7.63854946580985942742931688557, 8.091245920318994813920081512769

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