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Properties

Label 1-1575-1575.788-r0-0-0
Degree $1$
Conductor $1575$
Sign $0.907 - 0.419i$
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  + (0.743 + 0.669i)2-s + (0.104 + 0.994i)4-s + (−0.587 + 0.809i)8-s + (−0.309 − 0.951i)11-s + (−0.207 − 0.978i)13-s + (−0.978 + 0.207i)16-s + (−0.994 − 0.104i)17-s + (−0.913 + 0.406i)19-s + (0.406 − 0.913i)22-s + (0.951 − 0.309i)23-s + (0.5 − 0.866i)26-s + (−0.104 − 0.994i)29-s + (0.913 − 0.406i)31-s + (−0.866 − 0.5i)32-s + (−0.669 − 0.743i)34-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)2-s + (0.104 + 0.994i)4-s + (−0.587 + 0.809i)8-s + (−0.309 − 0.951i)11-s + (−0.207 − 0.978i)13-s + (−0.978 + 0.207i)16-s + (−0.994 − 0.104i)17-s + (−0.913 + 0.406i)19-s + (0.406 − 0.913i)22-s + (0.951 − 0.309i)23-s + (0.5 − 0.866i)26-s + (−0.104 − 0.994i)29-s + (0.913 − 0.406i)31-s + (−0.866 − 0.5i)32-s + (−0.669 − 0.743i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.907 - 0.419i$
Analytic conductor: \(7.31426\)
Root analytic conductor: \(7.31426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (788, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1575,\ (0:\ ),\ 0.907 - 0.419i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.622168594 - 0.3571189289i\)
\(L(\frac12)\) \(\approx\) \(1.622168594 - 0.3571189289i\)
\(L(1)\) \(\approx\) \(1.304399359 + 0.2971353934i\)
\(L(1)\) \(\approx\) \(1.304399359 + 0.2971353934i\)

Euler product

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

−20.633418099132591432314231619396, −19.82988327670401058933722066492, −19.35573587127765581483266521884, −18.46055755473475511943953876239, −17.736515799598817300978452631550, −16.82483930092685449783359061935, −15.83825079861887938949380246357, −15.06554987452434695393953697934, −14.62018218919568891490379502588, −13.57585704882160755190298927497, −13.0174670326827026634262188975, −12.32009774819986424473726055735, −11.441334988698009612523543246311, −10.875662911764493895398375247353, −9.959065656014202334053290599228, −9.276936700698466968557533460839, −8.407109940692691665001386216883, −6.850443699218153398549033545067, −6.77089271882648940778423187470, −5.430368125656082009842242834528, −4.60064987486496084636985748051, −4.15244045505127031538313339035, −2.86884213924977823982964539846, −2.19247587464489784918328616956, −1.25848158455641046727706770182, 0.46682625351185023991538600787, 2.28496363586671098463621745032, 2.96895329403691563273559784729, 3.97171692586819503116636293818, 4.75316476990531928508855157603, 5.68695040281443530055888751969, 6.27352833878305437717969986509, 7.16474387856421262019552820884, 8.15335245608517509562360298872, 8.52445490733368348139263046211, 9.65654696543951878014119210785, 10.83540889578936175581052572583, 11.32873217439654498639447968825, 12.41758280317812014326028892966, 13.09235246213735923180315361972, 13.5801899373656374868865601345, 14.56297231431318184303779928359, 15.20555367678474469783919445578, 15.82426873055318446068973128764, 16.65267344258959421206598448316, 17.31716772764983248507440441301, 18.00475366282715344706980603166, 18.97206787600555135731849947714, 19.77238708936980861195919125904, 20.79102242332259805724618046195

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