Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 601 $
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 + 0.277·3-s + 4-s + 5-s + 0.277·6-s + 2.55·7-s + 8-s − 2.92·9-s + 10-s − 4.94·11-s + 0.277·12-s + 4.28·13-s + 2.55·14-s + 0.277·15-s + 16-s − 3.28·17-s − 2.92·18-s − 3.94·19-s + 20-s + 0.708·21-s − 4.94·22-s − 6.03·23-s + 0.277·24-s + 25-s + 4.28·26-s − 1.64·27-s + 2.55·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.160·3-s + 0.5·4-s + 0.447·5-s + 0.113·6-s + 0.964·7-s + 0.353·8-s − 0.974·9-s + 0.316·10-s − 1.48·11-s + 0.0801·12-s + 1.18·13-s + 0.682·14-s + 0.0717·15-s + 0.250·16-s − 0.797·17-s − 0.688·18-s − 0.904·19-s + 0.223·20-s + 0.154·21-s − 1.05·22-s − 1.25·23-s + 0.0566·24-s + 0.200·25-s + 0.840·26-s − 0.316·27-s + 0.482·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6010\)    =    \(2 \cdot 5 \cdot 601\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6010} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6010,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;601\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;601\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
5 \( 1 - T \)
601 \( 1 + T \)
good3 \( 1 - 0.277T + 3T^{2} \)
7 \( 1 - 2.55T + 7T^{2} \)
11 \( 1 + 4.94T + 11T^{2} \)
13 \( 1 - 4.28T + 13T^{2} \)
17 \( 1 + 3.28T + 17T^{2} \)
19 \( 1 + 3.94T + 19T^{2} \)
23 \( 1 + 6.03T + 23T^{2} \)
29 \( 1 + 7.47T + 29T^{2} \)
31 \( 1 + 4.60T + 31T^{2} \)
37 \( 1 + 1.65T + 37T^{2} \)
41 \( 1 + 8.01T + 41T^{2} \)
43 \( 1 - 1.78T + 43T^{2} \)
47 \( 1 + 3.33T + 47T^{2} \)
53 \( 1 + 8.76T + 53T^{2} \)
59 \( 1 - 6.61T + 59T^{2} \)
61 \( 1 - 10.0T + 61T^{2} \)
67 \( 1 + 5.12T + 67T^{2} \)
71 \( 1 - 2.50T + 71T^{2} \)
73 \( 1 - 2.44T + 73T^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 - 1.46T + 83T^{2} \)
89 \( 1 + 7.59T + 89T^{2} \)
97 \( 1 + 0.655T + 97T^{2} \)
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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

−7.947263528892518377925914034697, −6.88419172350969598643468348087, −6.07108949316439655659271710877, −5.52571356182780720960619464759, −4.98165139566464138326695345229, −4.04835316357076535555108607253, −3.28013286061321824646870075533, −2.22068903692760228475007785831, −1.83211497999609680517250477100, 0, 1.83211497999609680517250477100, 2.22068903692760228475007785831, 3.28013286061321824646870075533, 4.04835316357076535555108607253, 4.98165139566464138326695345229, 5.52571356182780720960619464759, 6.07108949316439655659271710877, 6.88419172350969598643468348087, 7.947263528892518377925914034697

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