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Properties

Label 1-1575-1575.1102-r0-0-0
Degree $1$
Conductor $1575$
Sign $-0.663 - 0.748i$
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  + (−0.207 − 0.978i)2-s + (−0.913 + 0.406i)4-s + (0.587 + 0.809i)8-s + (0.309 − 0.951i)11-s + (0.743 + 0.669i)13-s + (0.669 − 0.743i)16-s + (0.406 − 0.913i)17-s + (−0.104 − 0.994i)19-s + (−0.994 − 0.104i)22-s + (−0.951 − 0.309i)23-s + (0.5 − 0.866i)26-s + (−0.913 + 0.406i)29-s + (0.104 + 0.994i)31-s + (−0.866 − 0.5i)32-s + (−0.978 − 0.207i)34-s + ⋯
L(s)  = 1  + (−0.207 − 0.978i)2-s + (−0.913 + 0.406i)4-s + (0.587 + 0.809i)8-s + (0.309 − 0.951i)11-s + (0.743 + 0.669i)13-s + (0.669 − 0.743i)16-s + (0.406 − 0.913i)17-s + (−0.104 − 0.994i)19-s + (−0.994 − 0.104i)22-s + (−0.951 − 0.309i)23-s + (0.5 − 0.866i)26-s + (−0.913 + 0.406i)29-s + (0.104 + 0.994i)31-s + (−0.866 − 0.5i)32-s + (−0.978 − 0.207i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.663 - 0.748i$
Analytic conductor: \(7.31426\)
Root analytic conductor: \(7.31426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1575,\ (0:\ ),\ -0.663 - 0.748i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4752622939 - 1.055971656i\)
\(L(\frac12)\) \(\approx\) \(0.4752622939 - 1.055971656i\)
\(L(1)\) \(\approx\) \(0.7572388169 - 0.4975735032i\)
\(L(1)\) \(\approx\) \(0.7572388169 - 0.4975735032i\)

Euler product

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

−20.61349527319959461508833915558, −20.089549319784192648921816522275, −18.91038586225676961010381940127, −18.58564807536374829206245423439, −17.545177120701293880074851079007, −17.14654582044334955466390007822, −16.32366829562745800635246755452, −15.41598524098543560817590089775, −15.01283008010180026697425854563, −14.18370152679294176881313450590, −13.36849808920213112634408094038, −12.64637425859221048246378048258, −11.82105455846092875179127869232, −10.52358058298833239452192970319, −10.03205696287570957752983008389, −9.19826609361614964641257727554, −8.20880793526169679378370558527, −7.76960491099084447455256967630, −6.818742432694954317965449587536, −5.89638177705148250500931261435, −5.47305339442765621137171213896, −4.09133389989890083014002422163, −3.79003705397907273591911406299, −2.091848712509057088964551028850, −1.095579297688505489606266301612, 0.54822555799601907427124274302, 1.54840064274056094192831020112, 2.59085646983168518699064521880, 3.43791975981326983183072384708, 4.19634207008891953781670679693, 5.15088090493635884932397592519, 6.090605219673017148330782808521, 7.158111194797761306603942207596, 8.12788957076954105647639949252, 8.98808364804230209695504992637, 9.36543544293268361208830524522, 10.51936851705413929557776161957, 11.12514926852080731642121371311, 11.75290591697286964781791144928, 12.51735660448035904607709081054, 13.549554388879778496239376948243, 13.88378721390081393479640815983, 14.737780452232364416623397207787, 16.11606362926408023266801050470, 16.426065237032532037439060374116, 17.48827985385804260149542667257, 18.21204126422308386105781752760, 18.8028483745816361100105636512, 19.50472361710882244116143351133, 20.23679971520099994197711236051

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