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Properties

Label 1-975-975.44-r0-0-0
Degree $1$
Conductor $975$
Sign $0.552 + 0.833i$
Analytic cond. $4.52788$
Root an. cond. $4.52788$
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  + (0.587 + 0.809i)2-s + (−0.309 + 0.951i)4-s − i·7-s + (−0.951 + 0.309i)8-s + (0.587 + 0.809i)11-s + (0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)17-s + (0.951 − 0.309i)19-s + (−0.309 + 0.951i)22-s + (0.809 − 0.587i)23-s + (0.951 + 0.309i)28-s + (−0.309 + 0.951i)29-s + (0.951 − 0.309i)31-s − i·32-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (−0.309 + 0.951i)4-s − i·7-s + (−0.951 + 0.309i)8-s + (0.587 + 0.809i)11-s + (0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s + (−0.309 − 0.951i)17-s + (0.951 − 0.309i)19-s + (−0.309 + 0.951i)22-s + (0.809 − 0.587i)23-s + (0.951 + 0.309i)28-s + (−0.309 + 0.951i)29-s + (0.951 − 0.309i)31-s − i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.552 + 0.833i$
Analytic conductor: \(4.52788\)
Root analytic conductor: \(4.52788\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 975,\ (0:\ ),\ 0.552 + 0.833i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.779012147 + 0.9547982813i\)
\(L(\frac12)\) \(\approx\) \(1.779012147 + 0.9547982813i\)
\(L(1)\) \(\approx\) \(1.326374816 + 0.5580971818i\)
\(L(1)\) \(\approx\) \(1.326374816 + 0.5580971818i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.587 + 0.809i)T \)
7 \( 1 - iT \)
11 \( 1 + (0.587 + 0.809i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.951 - 0.309i)T \)
23 \( 1 + (0.809 - 0.587i)T \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (0.951 - 0.309i)T \)
37 \( 1 + (-0.587 + 0.809i)T \)
41 \( 1 + (0.587 - 0.809i)T \)
43 \( 1 + T \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.587 + 0.809i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (-0.951 + 0.309i)T \)
71 \( 1 + (0.951 + 0.309i)T \)
73 \( 1 + (0.587 + 0.809i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (0.951 - 0.309i)T \)
89 \( 1 + (0.587 + 0.809i)T \)
97 \( 1 + (0.951 + 0.309i)T \)
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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

−21.37736675367057633557929935507, −21.21772979118792680631445368770, −19.97899420237358819487487058656, −19.29323611842826491190600487832, −18.776115794657361822548169151230, −17.88329598327219358524802513619, −16.96911697178899760171189847564, −15.72269449912468092779106511995, −15.248794510376853268478579344289, −14.26474070227320816650112548738, −13.64044130378802010700712036154, −12.67862480024916148155373123556, −12.000047569759531869990238777941, −11.30582508629655755574147117821, −10.541592110570936922254196067313, −9.37404990235144439526005044352, −8.961394122124515468799951424, −7.82026807627065984743562753915, −6.333467224782781058280217260957, −5.83979275998014743885433349800, −4.92133628980563439674085800006, −3.82472250507310676502370478721, −3.05203892276825340943755051843, −2.06926844410479661088660435906, −1.01702039760210240025142072762, 0.950880002611318061671407234378, 2.57691935105551432246447872200, 3.581589309968603247910042302804, 4.50780596182593002190949632420, 5.08496698692195441607077190100, 6.34091076415513314296055151614, 7.17019082806977674300712061066, 7.50397782755362958218923295884, 8.81841335507438177788318306046, 9.49905528122531909264657106915, 10.61530414250411210081817791396, 11.64717843996928320231682233148, 12.38357255492622826948984924196, 13.34549764273300263600214376181, 13.94385887657298105320217202784, 14.64063887206906449490813749271, 15.53797755247571625613807070621, 16.2725998050091813261791522654, 17.081314472114188258361959218588, 17.62670260279559080476483487069, 18.47355002268365566195674065320, 19.67553747587920054180236100883, 20.53066714494657171702811164923, 20.944178972324599870954801067772, 22.39843222160304018816982704476

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