Properties

Label 2-927-103.24-c0-0-0
Degree $2$
Conductor $927$
Sign $-0.488 - 0.872i$
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  + (−0.0922 + 0.995i)4-s + (−1.12 + 1.48i)7-s + (0.111 − 0.147i)13-s + (−0.982 − 0.183i)16-s + (−1.58 − 0.614i)19-s + (0.445 + 0.895i)25-s + (−1.37 − 1.25i)28-s + (−0.353 + 1.89i)31-s + (1.53 − 0.436i)37-s + (1.72 + 0.489i)43-s + (−0.678 − 2.38i)49-s + (0.136 + 0.124i)52-s + (0.876 + 1.75i)61-s + (0.273 − 0.961i)64-s + (1.07 − 0.811i)67-s + ⋯
L(s)  = 1  + (−0.0922 + 0.995i)4-s + (−1.12 + 1.48i)7-s + (0.111 − 0.147i)13-s + (−0.982 − 0.183i)16-s + (−1.58 − 0.614i)19-s + (0.445 + 0.895i)25-s + (−1.37 − 1.25i)28-s + (−0.353 + 1.89i)31-s + (1.53 − 0.436i)37-s + (1.72 + 0.489i)43-s + (−0.678 − 2.38i)49-s + (0.136 + 0.124i)52-s + (0.876 + 1.75i)61-s + (0.273 − 0.961i)64-s + (1.07 − 0.811i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.488 - 0.872i$
Analytic conductor: \(0.462633\)
Root analytic conductor: \(0.680171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 927,\ (\ :0),\ -0.488 - 0.872i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7291973053\)
\(L(\frac12)\) \(\approx\) \(0.7291973053\)
\(L(1)\) 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 \( 1 + (-0.273 - 0.961i)T \)
good2 \( 1 + (0.0922 - 0.995i)T^{2} \)
5 \( 1 + (-0.445 - 0.895i)T^{2} \)
7 \( 1 + (1.12 - 1.48i)T + (-0.273 - 0.961i)T^{2} \)
11 \( 1 + (-0.0922 + 0.995i)T^{2} \)
13 \( 1 + (-0.111 + 0.147i)T + (-0.273 - 0.961i)T^{2} \)
17 \( 1 + (-0.602 + 0.798i)T^{2} \)
19 \( 1 + (1.58 + 0.614i)T + (0.739 + 0.673i)T^{2} \)
23 \( 1 + (0.0922 + 0.995i)T^{2} \)
29 \( 1 + (0.445 + 0.895i)T^{2} \)
31 \( 1 + (0.353 - 1.89i)T + (-0.932 - 0.361i)T^{2} \)
37 \( 1 + (-1.53 + 0.436i)T + (0.850 - 0.526i)T^{2} \)
41 \( 1 + (0.445 - 0.895i)T^{2} \)
43 \( 1 + (-1.72 - 0.489i)T + (0.850 + 0.526i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.739 - 0.673i)T^{2} \)
59 \( 1 + (-0.273 + 0.961i)T^{2} \)
61 \( 1 + (-0.876 - 1.75i)T + (-0.602 + 0.798i)T^{2} \)
67 \( 1 + (-1.07 + 0.811i)T + (0.273 - 0.961i)T^{2} \)
71 \( 1 + (-0.445 + 0.895i)T^{2} \)
73 \( 1 + (-0.193 - 0.312i)T + (-0.445 + 0.895i)T^{2} \)
79 \( 1 + (1.02 + 0.634i)T + (0.445 + 0.895i)T^{2} \)
83 \( 1 + (-0.273 - 0.961i)T^{2} \)
89 \( 1 + (0.982 - 0.183i)T^{2} \)
97 \( 1 + (0.397 - 0.798i)T + (-0.602 - 0.798i)T^{2} \)
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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

−10.61366180410202475625573887740, −9.340158848233139566969619631835, −8.962216548517053040726434581131, −8.242407494113478271251583292975, −7.08220644627030508910414759183, −6.36012597472921577903690658791, −5.39999893382290379805976684738, −4.19771836520041208569802858146, −3.08523495069196569816987516025, −2.40955946440900425819670650871, 0.69477744725725714660389238710, 2.33782942037391667626744387024, 3.92191874196816794175233659053, 4.44227410276918015092772849765, 5.95761942930927881194040871363, 6.41920196461785327386061867770, 7.29413087510541500967249193006, 8.354556822625260402793819994732, 9.528193464393954004362739039493, 9.948683817294671859079186202838

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