Properties

Label 1-297-297.92-r1-0-0
Degree $1$
Conductor $297$
Sign $0.0354 - 0.999i$
Analytic cond. $31.9170$
Root an. cond. $31.9170$
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  + (0.882 − 0.469i)2-s + (0.559 − 0.829i)4-s + (−0.848 + 0.529i)5-s + (0.961 − 0.275i)7-s + (0.104 − 0.994i)8-s + (−0.5 + 0.866i)10-s + (0.990 − 0.139i)13-s + (0.719 − 0.694i)14-s + (−0.374 − 0.927i)16-s + (−0.669 − 0.743i)17-s + (−0.104 + 0.994i)19-s + (−0.0348 + 0.999i)20-s + (0.939 − 0.342i)23-s + (0.438 − 0.898i)25-s + (0.809 − 0.587i)26-s + ⋯
L(s)  = 1  + (0.882 − 0.469i)2-s + (0.559 − 0.829i)4-s + (−0.848 + 0.529i)5-s + (0.961 − 0.275i)7-s + (0.104 − 0.994i)8-s + (−0.5 + 0.866i)10-s + (0.990 − 0.139i)13-s + (0.719 − 0.694i)14-s + (−0.374 − 0.927i)16-s + (−0.669 − 0.743i)17-s + (−0.104 + 0.994i)19-s + (−0.0348 + 0.999i)20-s + (0.939 − 0.342i)23-s + (0.438 − 0.898i)25-s + (0.809 − 0.587i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0354 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0354 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.0354 - 0.999i$
Analytic conductor: \(31.9170\)
Root analytic conductor: \(31.9170\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 297,\ (1:\ ),\ 0.0354 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.321214456 - 2.240443124i\)
\(L(\frac12)\) \(\approx\) \(2.321214456 - 2.240443124i\)
\(L(1)\) \(\approx\) \(1.651783080 - 0.7106278350i\)
\(L(1)\) \(\approx\) \(1.651783080 - 0.7106278350i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.882 - 0.469i)T \)
5 \( 1 + (-0.848 + 0.529i)T \)
7 \( 1 + (0.961 - 0.275i)T \)
13 \( 1 + (0.990 - 0.139i)T \)
17 \( 1 + (-0.669 - 0.743i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (0.719 + 0.694i)T \)
31 \( 1 + (-0.615 - 0.788i)T \)
37 \( 1 + (-0.104 - 0.994i)T \)
41 \( 1 + (0.719 - 0.694i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (-0.559 - 0.829i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.997 + 0.0697i)T \)
61 \( 1 + (-0.615 + 0.788i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.669 - 0.743i)T \)
73 \( 1 + (0.913 + 0.406i)T \)
79 \( 1 + (-0.882 + 0.469i)T \)
83 \( 1 + (-0.990 - 0.139i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.848 + 0.529i)T \)
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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

−25.11624219846337283586731613064, −24.28391906170287718103677571746, −23.67527215317268201844040425303, −22.9907593582590217030858200469, −21.750686440507877807988322364282, −21.07365598607713114422902895086, −20.23283367695331333925442480596, −19.26114886599514299058490707443, −17.851017720352928257384648612217, −17.0849833717774383273441052035, −15.89547814209775471643059954862, −15.39309497894527026722566081509, −14.47086956080869125717671648927, −13.33999239451549925568742225451, −12.58432308120824573403257184221, −11.38968054925353614858444504028, −11.07006195630945645420684055660, −8.85643882193864486849992850591, −8.27539281033416712732382854472, −7.24065035634433202379219885671, −6.07393092572759716195493599588, −4.84387979498058175671581388960, −4.25157430696742342532832822612, −2.96025902326894038682345607691, −1.382958462548187035485702259315, 0.78357820325053214594333273739, 2.19840349942043607669147689991, 3.50834686055154082723417015508, 4.30084121759463547613337337224, 5.40706030916861501799399434224, 6.69061063041199659240434948003, 7.61735093918523435898733047929, 8.84822170317272521309643704571, 10.50013744981913913224755617036, 11.052446968052851845695272394407, 11.811017383706865412331393301782, 12.87758862624796771727713030960, 14.0117186446823003530748646019, 14.65684944099820516665695835569, 15.56390616824591443986868303130, 16.4133354636373729617343011622, 18.00662509402862611985981968885, 18.723266356932007766447589210876, 19.756375162351999586536720646391, 20.61663122494458449088773684818, 21.22757023871555368548899671150, 22.51972198327443531766208571772, 23.02487299130295827807742159316, 23.8653892584359963416224540365, 24.63845377399594528433409266750

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