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Properties

Label 2-927-103.56-c1-0-23
Degree $2$
Conductor $927$
Sign $0.658 + 0.752i$
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  + (0.711 − 1.23i)2-s + (−0.0133 − 0.0230i)4-s + (−0.396 − 0.686i)5-s + (−0.249 − 0.432i)7-s + 2.80·8-s − 1.12·10-s + (2.68 + 4.64i)11-s − 1.03·13-s − 0.710·14-s + (2.02 − 3.50i)16-s + (0.783 + 1.35i)17-s + (2.21 − 3.84i)19-s + (−0.0105 + 0.0183i)20-s + 7.63·22-s + 5.69·23-s + ⋯
L(s)  = 1  + (0.503 − 0.871i)2-s + (−0.00666 − 0.0115i)4-s + (−0.177 − 0.307i)5-s + (−0.0943 − 0.163i)7-s + 0.993·8-s − 0.356·10-s + (0.808 + 1.39i)11-s − 0.285·13-s − 0.189·14-s + (0.506 − 0.877i)16-s + (0.189 + 0.329i)17-s + (0.509 − 0.882i)19-s + (−0.00236 + 0.00409i)20-s + 1.62·22-s + 1.18·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.658 + 0.752i$
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: \((2,\ 927,\ (\ :1/2),\ 0.658 + 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14832 - 0.974877i\)
\(L(\frac12)\) \(\approx\) \(2.14832 - 0.974877i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.20 + 9.23i)T \)
good2 \( 1 + (-0.711 + 1.23i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.396 + 0.686i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.249 + 0.432i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.68 - 4.64i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.03T + 13T^{2} \)
17 \( 1 + (-0.783 - 1.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.21 + 3.84i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.69T + 23T^{2} \)
29 \( 1 + (1.25 - 2.17i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.48T + 31T^{2} \)
37 \( 1 + 6.62T + 37T^{2} \)
41 \( 1 + (-1.70 + 2.95i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.43 - 5.94i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.61 - 2.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.01 + 1.75i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.36 + 5.82i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 0.780T + 61T^{2} \)
67 \( 1 + (3.62 + 6.27i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.32 - 5.76i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 12.2T + 73T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 + (1.88 - 3.25i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 18.4T + 89T^{2} \)
97 \( 1 + (-0.821 + 1.42i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.07258409244367705363969737542, −9.349854657621851961197554328875, −8.343479303987057688277821762198, −7.21527924970067490194569096369, −6.80399529534113394090902109121, −5.11706276053697737797269484275, −4.51119477346941774394282478561, −3.57220991860126642423325279961, −2.48017512759741363980918509459, −1.29891482082331707621499111577, 1.26186639993896385767818439677, 3.03231465305660519331299312140, 3.98511092789597979885908457258, 5.24367749899270834364484826492, 5.82376128881650529227575294231, 6.75132611009719669840128072084, 7.36917012884191811085070297834, 8.368450008032501352994459276458, 9.206989215972737670228771802009, 10.24344196689616957446669238488

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