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Properties

Label 2-369-41.5-c1-0-6
Degree $2$
Conductor $369$
Sign $0.900 - 0.435i$
Analytic cond. $2.94647$
Root an. cond. $1.71653$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.522 + 0.718i)2-s + (0.374 − 1.15i)4-s + (−1.02 − 0.331i)5-s + (0.625 + 3.95i)7-s + (2.71 − 0.881i)8-s + (−0.294 − 0.906i)10-s + (1.14 − 2.23i)11-s + (4.25 + 0.673i)13-s + (−2.51 + 2.51i)14-s + (0.0903 + 0.0656i)16-s + (4.26 + 2.17i)17-s + (0.967 − 0.153i)19-s + (−0.763 + 1.05i)20-s + (2.20 − 0.349i)22-s + (1.78 − 1.29i)23-s + ⋯
L(s)  = 1  + (0.369 + 0.508i)2-s + (0.187 − 0.575i)4-s + (−0.456 − 0.148i)5-s + (0.236 + 1.49i)7-s + (0.959 − 0.311i)8-s + (−0.0931 − 0.286i)10-s + (0.344 − 0.675i)11-s + (1.17 + 0.186i)13-s + (−0.671 + 0.671i)14-s + (0.0225 + 0.0164i)16-s + (1.03 + 0.526i)17-s + (0.221 − 0.0351i)19-s + (−0.170 + 0.235i)20-s + (0.470 − 0.0744i)22-s + (0.372 − 0.270i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369\)    =    \(3^{2} \cdot 41\)
Sign: $0.900 - 0.435i$
Analytic conductor: \(2.94647\)
Root analytic conductor: \(1.71653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{369} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 369,\ (\ :1/2),\ 0.900 - 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73619 + 0.397686i\)
\(L(\frac12)\) \(\approx\) \(1.73619 + 0.397686i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
41 \( 1 + (6.18 + 1.67i)T \)
good2 \( 1 + (-0.522 - 0.718i)T + (-0.618 + 1.90i)T^{2} \)
5 \( 1 + (1.02 + 0.331i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.625 - 3.95i)T + (-6.65 + 2.16i)T^{2} \)
11 \( 1 + (-1.14 + 2.23i)T + (-6.46 - 8.89i)T^{2} \)
13 \( 1 + (-4.25 - 0.673i)T + (12.3 + 4.01i)T^{2} \)
17 \( 1 + (-4.26 - 2.17i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-0.967 + 0.153i)T + (18.0 - 5.87i)T^{2} \)
23 \( 1 + (-1.78 + 1.29i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.83 - 0.933i)T + (17.0 - 23.4i)T^{2} \)
31 \( 1 + (2.02 + 6.23i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.34 - 10.3i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (1.18 + 1.63i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + (1.55 - 9.83i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (6.91 - 3.52i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (7.77 - 5.64i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.263 - 0.362i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (1.59 + 3.12i)T + (-39.3 + 54.2i)T^{2} \)
71 \( 1 + (-6.39 + 12.5i)T + (-41.7 - 57.4i)T^{2} \)
73 \( 1 + 8.75iT - 73T^{2} \)
79 \( 1 + (1.93 + 1.93i)T + 79iT^{2} \)
83 \( 1 - 2.79T + 83T^{2} \)
89 \( 1 + (1.73 + 10.9i)T + (-84.6 + 27.5i)T^{2} \)
97 \( 1 + (-1.70 - 3.35i)T + (-57.0 + 78.4i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.53823688785256179196092251129, −10.71363299851634445118435867973, −9.504857563580746579690921350049, −8.577258795146184347644996708982, −7.77669707516522444184165285443, −6.24868539385155687485314277660, −5.87585868470135151765758811567, −4.76526364158498023896021742320, −3.36971249430829259270106356843, −1.58004084205194937915158383931, 1.48271089996431384494152781571, 3.45981921485619349618767498456, 3.88335553448802317543278899954, 5.15306217300582518089536313736, 6.92750854611952427305643996903, 7.46260945058945874273329288111, 8.336576289401241699750329505535, 9.755934198346683869692345480074, 10.74384123067253031401860687999, 11.29933010520055675783087720427

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