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Properties

Label 1-1033-1033.837-r0-0-0
Degree $1$
Conductor $1033$
Sign $0.317 - 0.948i$
Analytic cond. $4.79723$
Root an. cond. $4.79723$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 6-s + 7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + 15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 6-s + 7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + 15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $0.317 - 0.948i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (837, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ 0.317 - 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2686623617 - 0.1933631880i\)
\(L(\frac12)\) \(\approx\) \(0.2686623617 - 0.1933631880i\)
\(L(1)\) \(\approx\) \(0.5285702855 + 0.1114330141i\)
\(L(1)\) \(\approx\) \(0.5285702855 + 0.1114330141i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−21.46364768758195805664002234979, −21.01519585765667227976841012543, −20.217862211170787210040346226855, −19.67653154446319486678297775348, −18.59041611971985786337911949584, −17.70783899912390261752272325080, −17.09618267895686443040494338251, −16.515205974168932261224020880607, −15.5753693672704779581052544362, −14.83257503307332642663887935609, −13.607712256703704737283910543136, −12.72328265589616807493960436800, −11.91557720541011511517582944514, −11.32713895780646177836870852531, −10.550298110099311533777649798634, −9.92984468616168741754378921473, −8.68546746487326352339237659938, −8.39485790219745842868014900956, −7.51070998302870429010559172881, −5.68264022390894670055870728694, −5.1357999738318630282963069793, −4.08121040576451357508810138676, −3.6056001324572358753956144208, −2.17674639244875017653304038516, −0.9839846671733492973778540461, 0.21320370750699259475660648743, 1.78535130375949941106128041273, 2.48213454447783254965675421192, 4.52682235335344699331739162090, 4.8666126301177091158283522964, 6.19120133460625668004572346561, 6.899411180708288242928020642254, 7.48900480438076032310179710278, 8.068598917240712132867054484824, 9.145408259942071172466901569673, 10.24799062011299634249303512847, 11.16447757379467148642282379947, 11.5622861669034873079148805356, 12.784572502349084071690171847039, 13.70606935985970759470699888337, 14.60482800345349738939854055832, 14.93778094545117762848626137220, 16.06224124588078623833919562879, 16.83269589585582780513832808035, 17.72271834992561037658811814898, 18.263303190181773724823969115749, 18.60747580127662025228911474957, 19.66656332370507396223532222784, 20.25411138639721134414621479765, 21.76464121652114764096521110483

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