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Properties

Label 2-927-103.56-c1-0-15
Degree $2$
Conductor $927$
Sign $-0.243 - 0.969i$
Analytic cond. $7.40213$
Root an. cond. $2.72068$
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  + (−1.35 + 2.34i)2-s + (−2.67 − 4.62i)4-s + (1.61 + 2.80i)5-s + (−0.805 − 1.39i)7-s + 9.06·8-s − 8.77·10-s + (−0.123 − 0.213i)11-s + 4.75·13-s + 4.36·14-s + (−6.93 + 12.0i)16-s + (−2.01 − 3.48i)17-s + (2.96 − 5.14i)19-s + (8.65 − 14.9i)20-s + 0.667·22-s + 3.42·23-s + ⋯
L(s)  = 1  + (−0.958 + 1.65i)2-s + (−1.33 − 2.31i)4-s + (0.723 + 1.25i)5-s + (−0.304 − 0.527i)7-s + 3.20·8-s − 2.77·10-s + (−0.0371 − 0.0642i)11-s + 1.32·13-s + 1.16·14-s + (−1.73 + 3.00i)16-s + (−0.488 − 0.845i)17-s + (0.680 − 1.17i)19-s + (1.93 − 3.35i)20-s + 0.142·22-s + 0.713·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.243 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.243 - 0.969i$
Analytic conductor: \(7.40213\)
Root analytic conductor: \(2.72068\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 927,\ (\ :1/2),\ -0.243 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.640915 + 0.821596i\)
\(L(\frac12)\) \(\approx\) \(0.640915 + 0.821596i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 + (0.467 - 10.1i)T \)
good2 \( 1 + (1.35 - 2.34i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.61 - 2.80i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.805 + 1.39i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.123 + 0.213i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.75T + 13T^{2} \)
17 \( 1 + (2.01 + 3.48i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.96 + 5.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.42T + 23T^{2} \)
29 \( 1 + (-1.32 + 2.29i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.09T + 31T^{2} \)
37 \( 1 + 5.29T + 37T^{2} \)
41 \( 1 + (-2.89 + 5.01i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.936 + 1.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.41 - 5.91i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.46 - 9.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.26 - 5.65i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 + (-7.95 - 13.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.17 - 2.03i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 - 1.11T + 79T^{2} \)
83 \( 1 + (-7.39 + 12.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 17.0T + 89T^{2} \)
97 \( 1 + (-8.38 + 14.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.19347562627671802331526043847, −9.216204615670988244958523866381, −8.712039196437222711495772567275, −7.44959663061213514866281603307, −6.98221154807272449699891160868, −6.33115548312778244964518085086, −5.61841448479899191538006424782, −4.43023595705361561387922330354, −2.81347701504364438637311816837, −0.942697976865330312138930851317, 1.06907090818464945733190673796, 1.81881905247757871393654483525, 3.11463256263246657403635330765, 4.11100290605146637404067022993, 5.19740379328131544362757777558, 6.33345175904241894878833507307, 8.001073582121137169653498224421, 8.575013318287538229309495807593, 9.104598639697972865672377511025, 9.824930601251199549208937544452

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