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Properties

Label 1-1575-1575.1147-r0-0-0
Degree $1$
Conductor $1575$
Sign $0.201 + 0.979i$
Analytic cond. $7.31426$
Root an. cond. $7.31426$
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  + (0.406 + 0.913i)2-s + (−0.669 + 0.743i)4-s + (−0.951 − 0.309i)8-s + (0.913 − 0.406i)11-s + (−0.406 + 0.913i)13-s + (−0.104 − 0.994i)16-s + (−0.951 − 0.309i)17-s + (0.309 − 0.951i)19-s + (0.743 + 0.669i)22-s + (0.994 + 0.104i)23-s − 26-s + (0.978 + 0.207i)29-s + (0.978 − 0.207i)31-s + (0.866 − 0.5i)32-s + (−0.104 − 0.994i)34-s + ⋯
L(s)  = 1  + (0.406 + 0.913i)2-s + (−0.669 + 0.743i)4-s + (−0.951 − 0.309i)8-s + (0.913 − 0.406i)11-s + (−0.406 + 0.913i)13-s + (−0.104 − 0.994i)16-s + (−0.951 − 0.309i)17-s + (0.309 − 0.951i)19-s + (0.743 + 0.669i)22-s + (0.994 + 0.104i)23-s − 26-s + (0.978 + 0.207i)29-s + (0.978 − 0.207i)31-s + (0.866 − 0.5i)32-s + (−0.104 − 0.994i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.201 + 0.979i$
Analytic conductor: \(7.31426\)
Root analytic conductor: \(7.31426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1575,\ (0:\ ),\ 0.201 + 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.391698376 + 1.135041499i\)
\(L(\frac12)\) \(\approx\) \(1.391698376 + 1.135041499i\)
\(L(1)\) \(\approx\) \(1.097468531 + 0.6023702971i\)
\(L(1)\) \(\approx\) \(1.097468531 + 0.6023702971i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.406 + 0.913i)T \)
11 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (-0.406 + 0.913i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.994 + 0.104i)T \)
29 \( 1 + (0.978 + 0.207i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (-0.913 - 0.406i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (-0.207 + 0.978i)T \)
53 \( 1 + (0.951 - 0.309i)T \)
59 \( 1 + (0.913 + 0.406i)T \)
61 \( 1 + (-0.913 + 0.406i)T \)
67 \( 1 + (-0.207 - 0.978i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (-0.743 + 0.669i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (-0.207 + 0.978i)T \)
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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

−20.33153442810237796285534650785, −19.70275199447975932794986424421, −19.12075333846235366254794330663, −18.19555431814253134129661282649, −17.49562480721203213605370368923, −16.86181571275771624096673420110, −15.47067508775447879663775277293, −15.07762073295604180977608234173, −14.16840563842989032778818032432, −13.51063628908287954314430022099, −12.63662668736263035285212761854, −12.06119103726049521492410126414, −11.358044891759806431485444912527, −10.336286028275394711047227800890, −9.9633214307456967513448148941, −8.90588151395096534804780855154, −8.28094314253713173644562588523, −6.97931907611685591374890339285, −6.20331469477234248386404699730, −5.19715350084579975703071361202, −4.49327366711971383458027786389, −3.59585517310076421939250678164, −2.77224805986771907643264988802, −1.791444339042528001485216442624, −0.83393802788729570249368997376, 0.85201408907768847748559056449, 2.38886937583667732098308591210, 3.33282287993605720734674722910, 4.38057384642142146162703529442, 4.85092158154791497445902814629, 5.957051608649449997487319645170, 6.83803761880585829767285267069, 7.11647941054367944816740522842, 8.40603854694603683363736298863, 9.022328206626443282134571456950, 9.58776770498762364884610083352, 10.96778679735930501922316864255, 11.72530739665359188099472834478, 12.41383399668363671400237097054, 13.48030011689274046269251780969, 13.88929632768595549635422581816, 14.671548774325637121916848978013, 15.47673693219435449158256649660, 16.101290526160183460244720920716, 16.94289943302670185803465317103, 17.4607998062939841589914543629, 18.21058306887009995483810723496, 19.2602036128931795357889083592, 19.7184180775728806440711977786, 21.030395970409379860133246440947

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