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Properties

Label 2-927-103.46-c1-0-31
Degree $2$
Conductor $927$
Sign $0.976 + 0.215i$
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  + (0.609 + 1.05i)2-s + (0.256 − 0.445i)4-s + (1.48 − 2.57i)5-s + (0.0501 − 0.0869i)7-s + 3.06·8-s + 3.62·10-s + (−0.856 + 1.48i)11-s + 0.822·13-s + 0.122·14-s + (1.35 + 2.34i)16-s + (1.04 − 1.81i)17-s + (−0.333 − 0.577i)19-s + (−0.763 − 1.32i)20-s − 2.08·22-s − 4.28·23-s + ⋯
L(s)  = 1  + (0.431 + 0.746i)2-s + (0.128 − 0.222i)4-s + (0.664 − 1.15i)5-s + (0.0189 − 0.0328i)7-s + 1.08·8-s + 1.14·10-s + (−0.258 + 0.447i)11-s + 0.228·13-s + 0.0327·14-s + (0.338 + 0.586i)16-s + (0.254 − 0.441i)17-s + (−0.0765 − 0.132i)19-s + (−0.170 − 0.295i)20-s − 0.445·22-s − 0.893·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.976 + 0.215i$
Analytic conductor: \(7.40213\)
Root analytic conductor: \(2.72068\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 927,\ (\ :1/2),\ 0.976 + 0.215i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.47995 - 0.270179i\)
\(L(\frac12)\) \(\approx\) \(2.47995 - 0.270179i\)
\(L(\frac{3}{2})\) 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 \( 1 + (10.1 - 0.759i)T \)
good2 \( 1 + (-0.609 - 1.05i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.48 + 2.57i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.0501 + 0.0869i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.856 - 1.48i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.822T + 13T^{2} \)
17 \( 1 + (-1.04 + 1.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.333 + 0.577i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.28T + 23T^{2} \)
29 \( 1 + (2.93 + 5.08i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.90T + 31T^{2} \)
37 \( 1 - 9.62T + 37T^{2} \)
41 \( 1 + (3.86 + 6.69i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.963 - 1.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.40 - 2.43i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.851 - 1.47i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.84 + 4.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 12.5T + 61T^{2} \)
67 \( 1 + (3.36 - 5.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.27 - 7.41i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.38T + 73T^{2} \)
79 \( 1 - 9.93T + 79T^{2} \)
83 \( 1 + (-6.81 - 11.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.260T + 89T^{2} \)
97 \( 1 + (1.23 + 2.13i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.868090143134162693617800544014, −9.300235255796473502092235143963, −8.160386409277782894272580060007, −7.51270326228815315344280535610, −6.33412490228227827317901279488, −5.75130818185706486026739224517, −4.89062525834386807842302369549, −4.21973323655551036462359116294, −2.34190099497865828165253767696, −1.14189551807634805099125703250, 1.71238074228950255174106910095, 2.75785016482581409096059771181, 3.43731696697346600618599904016, 4.55977601381832708860593278541, 5.86424498081151126441115423718, 6.55229510713518802768260906230, 7.57318615192201335804805401335, 8.319494492783963095700146001969, 9.607768894360401525760795283826, 10.44273418894648152845336336212

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