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Properties

Label 1-1032-1032.515-r1-0-0
Degree $1$
Conductor $1032$
Sign $1$
Analytic cond. $110.903$
Root an. cond. $110.903$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 7-s − 11-s − 13-s − 17-s − 19-s + 23-s + 25-s − 29-s − 31-s − 35-s + 37-s − 41-s + 47-s + 49-s + 53-s + 55-s − 59-s + 61-s + 65-s + 67-s − 71-s − 73-s − 77-s − 79-s − 83-s + 85-s + ⋯
L(s)  = 1  − 5-s + 7-s − 11-s − 13-s − 17-s − 19-s + 23-s + 25-s − 29-s − 31-s − 35-s + 37-s − 41-s + 47-s + 49-s + 53-s + 55-s − 59-s + 61-s + 65-s + 67-s − 71-s − 73-s − 77-s − 79-s − 83-s + 85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1032 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1032 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1032\)    =    \(2^{3} \cdot 3 \cdot 43\)
Sign: $1$
Analytic conductor: \(110.903\)
Root analytic conductor: \(110.903\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1032} (515, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1032,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.000665712\)
\(L(\frac12)\) \(\approx\) \(1.000665712\)
\(L(1)\) \(\approx\) \(0.7823480619\)
\(L(1)\) \(\approx\) \(0.7823480619\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
43 \( 1 \)
good5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 - T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 + T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.41816851960962432136810726267, −20.421621148682226951481161862372, −20.00143367104439169125611555469, −18.92547072054982617104835704739, −18.4347910769349083291383097157, −17.39679548045237053035797420861, −16.77867192774777142725942760565, −15.7345335226795635390287524277, −14.97414170259729969292355854722, −14.66402934164398150567914334125, −13.28905126440998159893992003365, −12.69331054791544779098085747567, −11.69058557084681746985504721822, −11.04061328868341601794632365823, −10.42274327043209414994004211734, −9.07711595636141395655989937278, −8.37426552596979803626854703069, −7.52826342465664866923231536994, −7.00329569884249073758383828469, −5.53685459527451659742056999947, −4.74082478960169313423732023238, −4.08883465996718451299868257504, −2.79584777994925078025518421931, −1.93644440967103022875560003267, −0.4439950271306832639166165537, 0.4439950271306832639166165537, 1.93644440967103022875560003267, 2.79584777994925078025518421931, 4.08883465996718451299868257504, 4.74082478960169313423732023238, 5.53685459527451659742056999947, 7.00329569884249073758383828469, 7.52826342465664866923231536994, 8.37426552596979803626854703069, 9.07711595636141395655989937278, 10.42274327043209414994004211734, 11.04061328868341601794632365823, 11.69058557084681746985504721822, 12.69331054791544779098085747567, 13.28905126440998159893992003365, 14.66402934164398150567914334125, 14.97414170259729969292355854722, 15.7345335226795635390287524277, 16.77867192774777142725942760565, 17.39679548045237053035797420861, 18.4347910769349083291383097157, 18.92547072054982617104835704739, 20.00143367104439169125611555469, 20.421621148682226951481161862372, 21.41816851960962432136810726267

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