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Properties

Label 2-927-103.3-c0-0-0
Degree $2$
Conductor $927$
Sign $-0.0230 - 0.999i$
Analytic cond. $0.462633$
Root an. cond. $0.680171$
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  + (−0.445 + 0.895i)4-s + (−0.0505 + 0.544i)7-s + (−0.0822 + 0.887i)13-s + (−0.602 − 0.798i)16-s + (−0.510 + 1.79i)19-s + (0.739 − 0.673i)25-s + (−0.465 − 0.288i)28-s + (−0.293 + 0.221i)31-s + (−0.365 + 1.95i)37-s + (−0.247 − 1.32i)43-s + (0.688 + 0.128i)49-s + (−0.757 − 0.469i)52-s + (0.890 − 0.811i)61-s + (0.982 − 0.183i)64-s + (1.04 − 0.0971i)67-s + ⋯
L(s)  = 1  + (−0.445 + 0.895i)4-s + (−0.0505 + 0.544i)7-s + (−0.0822 + 0.887i)13-s + (−0.602 − 0.798i)16-s + (−0.510 + 1.79i)19-s + (0.739 − 0.673i)25-s + (−0.465 − 0.288i)28-s + (−0.293 + 0.221i)31-s + (−0.365 + 1.95i)37-s + (−0.247 − 1.32i)43-s + (0.688 + 0.128i)49-s + (−0.757 − 0.469i)52-s + (0.890 − 0.811i)61-s + (0.982 − 0.183i)64-s + (1.04 − 0.0971i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0230 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0230 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.0230 - 0.999i$
Analytic conductor: \(0.462633\)
Root analytic conductor: \(0.680171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 927,\ (\ :0),\ -0.0230 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8205455629\)
\(L(\frac12)\) \(\approx\) \(0.8205455629\)
\(L(1)\) 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 \( 1 + (-0.982 - 0.183i)T \)
good2 \( 1 + (0.445 - 0.895i)T^{2} \)
5 \( 1 + (-0.739 + 0.673i)T^{2} \)
7 \( 1 + (0.0505 - 0.544i)T + (-0.982 - 0.183i)T^{2} \)
11 \( 1 + (-0.445 + 0.895i)T^{2} \)
13 \( 1 + (0.0822 - 0.887i)T + (-0.982 - 0.183i)T^{2} \)
17 \( 1 + (0.0922 - 0.995i)T^{2} \)
19 \( 1 + (0.510 - 1.79i)T + (-0.850 - 0.526i)T^{2} \)
23 \( 1 + (0.445 + 0.895i)T^{2} \)
29 \( 1 + (0.739 - 0.673i)T^{2} \)
31 \( 1 + (0.293 - 0.221i)T + (0.273 - 0.961i)T^{2} \)
37 \( 1 + (0.365 - 1.95i)T + (-0.932 - 0.361i)T^{2} \)
41 \( 1 + (0.739 + 0.673i)T^{2} \)
43 \( 1 + (0.247 + 1.32i)T + (-0.932 + 0.361i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.850 + 0.526i)T^{2} \)
59 \( 1 + (-0.982 + 0.183i)T^{2} \)
61 \( 1 + (-0.890 + 0.811i)T + (0.0922 - 0.995i)T^{2} \)
67 \( 1 + (-1.04 + 0.0971i)T + (0.982 - 0.183i)T^{2} \)
71 \( 1 + (-0.739 - 0.673i)T^{2} \)
73 \( 1 + (-0.576 + 1.48i)T + (-0.739 - 0.673i)T^{2} \)
79 \( 1 + (0.172 - 0.0666i)T + (0.739 - 0.673i)T^{2} \)
83 \( 1 + (-0.982 - 0.183i)T^{2} \)
89 \( 1 + (0.602 - 0.798i)T^{2} \)
97 \( 1 + (1.09 + 0.995i)T + (0.0922 + 0.995i)T^{2} \)
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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

−10.36820208196635420682429620372, −9.572113009917442048645491942628, −8.612574285847447747526611204482, −8.243069535494411416341243316913, −7.13054683282773517819313310557, −6.28649322158564817713353142140, −5.12964210663648410482514118707, −4.17560563135876479333559101479, −3.27544706221619071111870980036, −2.01060155150798282923743455533, 0.836604648564770967463425890609, 2.47198849379318853711675659031, 3.86461256116506590996766697421, 4.87569636785016182320612354413, 5.58317105411469787572625291945, 6.66358372443232912271682564736, 7.41424291509970737579663635580, 8.602489354017380199805228936107, 9.256625746952853509669635657922, 10.09691614792657793698508351192

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