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Properties

Label 1-975-975.302-r0-0-0
Degree $1$
Conductor $975$
Sign $-0.923 - 0.383i$
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  + (0.743 − 0.669i)2-s + (0.104 − 0.994i)4-s + (0.866 − 0.5i)7-s + (−0.587 − 0.809i)8-s + (−0.669 − 0.743i)11-s + (0.309 − 0.951i)14-s + (−0.978 − 0.207i)16-s + (0.406 − 0.913i)17-s + (−0.913 − 0.406i)19-s + (−0.994 − 0.104i)22-s + (−0.743 + 0.669i)23-s + (−0.406 − 0.913i)28-s + (0.913 − 0.406i)29-s + (−0.809 + 0.587i)31-s + (−0.866 + 0.5i)32-s + ⋯
L(s)  = 1  + (0.743 − 0.669i)2-s + (0.104 − 0.994i)4-s + (0.866 − 0.5i)7-s + (−0.587 − 0.809i)8-s + (−0.669 − 0.743i)11-s + (0.309 − 0.951i)14-s + (−0.978 − 0.207i)16-s + (0.406 − 0.913i)17-s + (−0.913 − 0.406i)19-s + (−0.994 − 0.104i)22-s + (−0.743 + 0.669i)23-s + (−0.406 − 0.913i)28-s + (0.913 − 0.406i)29-s + (−0.809 + 0.587i)31-s + (−0.866 + 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.923 - 0.383i$
Analytic conductor: \(4.52788\)
Root analytic conductor: \(4.52788\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (302, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 975,\ (0:\ ),\ -0.923 - 0.383i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3808968604 - 1.909031367i\)
\(L(\frac12)\) \(\approx\) \(0.3808968604 - 1.909031367i\)
\(L(1)\) \(\approx\) \(1.138629691 - 0.9601128343i\)
\(L(1)\) \(\approx\) \(1.138629691 - 0.9601128343i\)

Euler product

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

−21.97145342423081038171862904704, −21.46108525562403597801171313418, −20.73802046167102882795994838836, −20.00215751678933583897399052646, −18.65827679652267009439782808270, −18.034468648380720121504408395, −17.23009744896853207320059797066, −16.507147995683409237544528504118, −15.51598134479184746199700979625, −14.883431758295771844812339388181, −14.40218555679724206587936405530, −13.33300706569821965918372441300, −12.50894834819061275029739444764, −12.00591044161841102211303381645, −10.92315963740639469424182760482, −10.04925625766231127689913363846, −8.62885535999971166427183089755, −8.1833313533826973617465248726, −7.32228371743297030615359920161, −6.27773442442240711609052796330, −5.511549872059550206543792540623, −4.65387068427781444369393617432, −3.92712000228191241360110399291, −2.61438269564462037486217452555, −1.82216008463993838019247771692, 0.6137895892693644923819027810, 1.791166302468831129456642317856, 2.76061468123455406597064941545, 3.739064361542934611613250458858, 4.68328424589116114023211605650, 5.366422897640165470832141182982, 6.321043304187378815095875661490, 7.42218371511166316470897101232, 8.32550930420366371891326707779, 9.425575213590776060034920540062, 10.3668261927374186553124430241, 11.06129878194987093849433707983, 11.639775274196449508718742474084, 12.64273585504633721049880702608, 13.467761262650464882130853343674, 14.11095769775887416343416342888, 14.75293298466431052139057416034, 15.78206841586361773493451213429, 16.443467158940095560787114900950, 17.761680137987402422331331133899, 18.2372299940936441756288318736, 19.31847420295861973485835917470, 19.864009889737515154730426069854, 20.86711973825537912453362943001, 21.28171249421456058157194815799

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