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Properties

Label 1-1032-1032.773-r0-0-0
Degree $1$
Conductor $1032$
Sign $1$
Analytic cond. $4.79258$
Root an. cond. $4.79258$
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

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

Invariants

Degree: \(1\)
Conductor: \(1032\)    =    \(2^{3} \cdot 3 \cdot 43\)
Sign: $1$
Analytic conductor: \(4.79258\)
Root analytic conductor: \(4.79258\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1032} (773, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1032,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8650594054\)
\(L(\frac12)\) \(\approx\) \(0.8650594054\)
\(L(1)\) \(\approx\) \(0.7772472199\)
\(L(1)\) \(\approx\) \(0.7772472199\)

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

−21.98762405616334812546735880521, −20.49982048151135812219127178543, −19.794790851020972113831467426347, −19.516200456231472633273226203014, −18.60385568680036254346103258629, −17.65989343831352273438006950957, −16.68091806319996478815048801536, −16.15578455060999778359631642180, −15.30410858949227584604459639073, −14.64349764914391919102928528807, −13.616326263235294971337269032484, −12.788259457121924833889646529704, −11.8352918230300261703701709057, −11.56038378211481335327119163205, −10.22593912124172235562461828251, −9.52798112798632142793722844168, −8.70570903024338790985425577944, −7.66061642010509180706178945241, −6.930230103477330682754527896195, −6.18392786483301457122084168954, −4.91565724049318837118752477394, −4.00728602463093897200692681413, −3.30618306361140722779565675680, −2.20262167915976398623733898199, −0.644007791625978482323228385778, 0.644007791625978482323228385778, 2.20262167915976398623733898199, 3.30618306361140722779565675680, 4.00728602463093897200692681413, 4.91565724049318837118752477394, 6.18392786483301457122084168954, 6.930230103477330682754527896195, 7.66061642010509180706178945241, 8.70570903024338790985425577944, 9.52798112798632142793722844168, 10.22593912124172235562461828251, 11.56038378211481335327119163205, 11.8352918230300261703701709057, 12.788259457121924833889646529704, 13.616326263235294971337269032484, 14.64349764914391919102928528807, 15.30410858949227584604459639073, 16.15578455060999778359631642180, 16.68091806319996478815048801536, 17.65989343831352273438006950957, 18.60385568680036254346103258629, 19.516200456231472633273226203014, 19.794790851020972113831467426347, 20.49982048151135812219127178543, 21.98762405616334812546735880521

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