Properties

Degree 1
Conductor $ 7 \cdot 11 \cdot 13 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 12-s − 15-s + 16-s + 17-s + 18-s − 19-s + 20-s + 23-s − 24-s + 25-s − 27-s − 29-s − 30-s + 31-s + 32-s + 34-s + 36-s − 37-s − 38-s + ⋯
L(s,χ)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 12-s − 15-s + 16-s + 17-s + 18-s − 19-s + 20-s + 23-s − 24-s + 25-s − 27-s − 29-s − 30-s + 31-s + 32-s + 34-s + 36-s − 37-s − 38-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1001\)    =    \(7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{1001} (1000, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 1001,\ (0:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.769093905$
$L(\frac12,\chi)$  $\approx$  $2.769093905$
$L(\chi,1)$  $\approx$  1.841773960
$L(1,\chi)$  $\approx$  1.841773960

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−21.74265820940144991375610658330, −21.07973176848511201735198743885, −20.61231707892567028872444756086, −19.19138160772353630767465267719, −18.59458658508285972006357527853, −17.33777579941455901617602829777, −16.9968444981433234078000112948, −16.235352219805653398217336415453, −15.202821695454490413486338059747, −14.56071080478199112716861071539, −13.46071230264861468027506877627, −13.016851790964565505291688841680, −12.15021519817850063502862935162, −11.43215835184530768689945169027, −10.40055900696255424945177913962, −10.092240638085773541022174031013, −8.701009115658206880488133545666, −7.2945821673295489930636741938, −6.65564409224680009661348867643, −5.78653453316782703632001445914, −5.27498182371009353634645780847, −4.401062626074685193700047697490, −3.27628639458399689037544550932, −2.09408684144974849212636822918, −1.19827459870646569310156473869, 1.19827459870646569310156473869, 2.09408684144974849212636822918, 3.27628639458399689037544550932, 4.401062626074685193700047697490, 5.27498182371009353634645780847, 5.78653453316782703632001445914, 6.65564409224680009661348867643, 7.2945821673295489930636741938, 8.701009115658206880488133545666, 10.092240638085773541022174031013, 10.40055900696255424945177913962, 11.43215835184530768689945169027, 12.15021519817850063502862935162, 13.016851790964565505291688841680, 13.46071230264861468027506877627, 14.56071080478199112716861071539, 15.202821695454490413486338059747, 16.235352219805653398217336415453, 16.9968444981433234078000112948, 17.33777579941455901617602829777, 18.59458658508285972006357527853, 19.19138160772353630767465267719, 20.61231707892567028872444756086, 21.07973176848511201735198743885, 21.74265820940144991375610658330

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