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Properties

Label 1-1575-1575.103-r0-0-0
Degree $1$
Conductor $1575$
Sign $-0.981 - 0.193i$
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  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (0.951 + 0.309i)8-s + (−0.104 + 0.994i)11-s + (0.406 − 0.913i)13-s + (−0.809 + 0.587i)16-s + (−0.743 + 0.669i)17-s + (−0.978 + 0.207i)19-s + (−0.743 − 0.669i)22-s + (0.406 + 0.913i)23-s + (0.5 + 0.866i)26-s + (0.978 + 0.207i)29-s + (−0.309 + 0.951i)31-s − i·32-s + (−0.104 − 0.994i)34-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (0.951 + 0.309i)8-s + (−0.104 + 0.994i)11-s + (0.406 − 0.913i)13-s + (−0.809 + 0.587i)16-s + (−0.743 + 0.669i)17-s + (−0.978 + 0.207i)19-s + (−0.743 − 0.669i)22-s + (0.406 + 0.913i)23-s + (0.5 + 0.866i)26-s + (0.978 + 0.207i)29-s + (−0.309 + 0.951i)31-s − i·32-s + (−0.104 − 0.994i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.981 - 0.193i$
Analytic conductor: \(7.31426\)
Root analytic conductor: \(7.31426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1575,\ (0:\ ),\ -0.981 - 0.193i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03899590292 + 0.3999484837i\)
\(L(\frac12)\) \(\approx\) \(-0.03899590292 + 0.3999484837i\)
\(L(1)\) \(\approx\) \(0.5831271578 + 0.2833129712i\)
\(L(1)\) \(\approx\) \(0.5831271578 + 0.2833129712i\)

Euler product

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

−20.09525048593901093528812599425, −19.291672480330988419973802153888, −18.651944874632492799080875431172, −18.19196482187787487143712769075, −17.08184228570216296414267333866, −16.62088884584663244903036910040, −15.88085839772036360678237232698, −14.83586779347697385538271378721, −13.71492391176637413282018441755, −13.36950969765093924989495250641, −12.41941868383370513132650882227, −11.4337573186357884982147676445, −11.160575495792293659730907369842, −10.22224795885041722741189283505, −9.37221741251009519262371130318, −8.586115265610102386188836390776, −8.16197586117339748336465630224, −6.86625466692006099855294932275, −6.315780731347204716307248560771, −4.83275703321903827676283047053, −4.21147685252263313723405537370, −3.143134724089803787692323628515, −2.40737404416909007097134240453, −1.3766481365663023909552640524, −0.18899505178462519346288565936, 1.326981576451481320100954411810, 2.17761669683034620841221908505, 3.571760580807134668376354007065, 4.62266974761558320864317979235, 5.3356080868480042901276554496, 6.299039994257501604710069933118, 6.95052223857246824055517918831, 7.81968996878070400781257695893, 8.55346687797418836516855510933, 9.22277922253377789196877616481, 10.41475582754840465493863649637, 10.48482679094938465431373918527, 11.73836453006789252139049157496, 12.866807663759512081913195778753, 13.3316281101881714729265933078, 14.472186300069187611857217299345, 15.076508846068396683215780153677, 15.61988917570234629977105923275, 16.408450650124906485375126630158, 17.44261645209074998889441975777, 17.68054059670085466955619277929, 18.441116371717139018266979456810, 19.52652883045538509663759996393, 19.80696218317213042085257723232, 20.7803598149381473663616234345

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