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 + 1.75·3-s + 4-s + 5-s + 1.75·6-s − 0.115·7-s + 8-s + 0.0923·9-s + 10-s − 1.05·11-s + 1.75·12-s − 5.92·13-s − 0.115·14-s + 1.75·15-s + 16-s − 7.65·17-s + 0.0923·18-s − 6.01·19-s + 20-s − 0.203·21-s − 1.05·22-s − 0.155·23-s + 1.75·24-s + 25-s − 5.92·26-s − 5.11·27-s − 0.115·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.01·3-s + 0.5·4-s + 0.447·5-s + 0.717·6-s − 0.0436·7-s + 0.353·8-s + 0.0307·9-s + 0.316·10-s − 0.319·11-s + 0.507·12-s − 1.64·13-s − 0.0308·14-s + 0.454·15-s + 0.250·16-s − 1.85·17-s + 0.0217·18-s − 1.37·19-s + 0.223·20-s − 0.0443·21-s − 0.225·22-s − 0.0325·23-s + 0.358·24-s + 0.200·25-s − 1.16·26-s − 0.984·27-s − 0.0218·28-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 \( 1 - 1.75T + 3T^{2} \)
7 \( 1 + 0.115T + 7T^{2} \)
11 \( 1 + 1.05T + 11T^{2} \)
13 \( 1 + 5.92T + 13T^{2} \)
17 \( 1 + 7.65T + 17T^{2} \)
19 \( 1 + 6.01T + 19T^{2} \)
23 \( 1 + 0.155T + 23T^{2} \)
29 \( 1 - 3.22T + 29T^{2} \)
31 \( 1 + 3.14T + 31T^{2} \)
37 \( 1 - 6.44T + 37T^{2} \)
41 \( 1 + 0.499T + 41T^{2} \)
43 \( 1 + 3.20T + 43T^{2} \)
47 \( 1 - 5.42T + 47T^{2} \)
53 \( 1 - 4.44T + 53T^{2} \)
59 \( 1 - 2.13T + 59T^{2} \)
61 \( 1 - 5.01T + 61T^{2} \)
67 \( 1 + 15.5T + 67T^{2} \)
71 \( 1 - 8.80T + 71T^{2} \)
73 \( 1 + 11.1T + 73T^{2} \)
79 \( 1 + 2.78T + 79T^{2} \)
83 \( 1 - 5.34T + 83T^{2} \)
89 \( 1 + 3.18T + 89T^{2} \)
97 \( 1 + 7.38T + 97T^{2} \)
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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

−7.68765571410023164258158444395, −6.96907413087378252342665789354, −6.35463046959182833116124147182, −5.50316559213606439791850789078, −4.62110661450077313064828551564, −4.18980089900586417053728365336, −3.02047656391385184105814670021, −2.38948869987546424625466755598, −2.00171943079084645692501644078, 0, 2.00171943079084645692501644078, 2.38948869987546424625466755598, 3.02047656391385184105814670021, 4.18980089900586417053728365336, 4.62110661450077313064828551564, 5.50316559213606439791850789078, 6.35463046959182833116124147182, 6.96907413087378252342665789354, 7.68765571410023164258158444395

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