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Properties

Label 2-927-103.10-c0-0-0
Degree $2$
Conductor $927$
Sign $0.976 + 0.214i$
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  + (0.850 − 0.526i)4-s + (−0.890 + 0.811i)7-s + (1.25 − 1.14i)13-s + (0.445 − 0.895i)16-s + (1.18 + 1.56i)19-s + (0.932 − 0.361i)25-s + (−0.329 + 1.15i)28-s + (−1.78 − 0.887i)31-s + (−1.34 − 0.124i)37-s + (−0.719 + 0.0666i)43-s + (0.0417 − 0.450i)49-s + (0.465 − 1.63i)52-s + (−0.831 + 0.322i)61-s + (−0.0922 − 0.995i)64-s + (−1.29 + 1.42i)67-s + ⋯
L(s)  = 1  + (0.850 − 0.526i)4-s + (−0.890 + 0.811i)7-s + (1.25 − 1.14i)13-s + (0.445 − 0.895i)16-s + (1.18 + 1.56i)19-s + (0.932 − 0.361i)25-s + (−0.329 + 1.15i)28-s + (−1.78 − 0.887i)31-s + (−1.34 − 0.124i)37-s + (−0.719 + 0.0666i)43-s + (0.0417 − 0.450i)49-s + (0.465 − 1.63i)52-s + (−0.831 + 0.322i)61-s + (−0.0922 − 0.995i)64-s + (−1.29 + 1.42i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.976 + 0.214i$
Analytic conductor: \(0.462633\)
Root analytic conductor: \(0.680171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 927,\ (\ :0),\ 0.976 + 0.214i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.169753331\)
\(L(\frac12)\) \(\approx\) \(1.169753331\)
\(L(1)\) 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 \( 1 + (0.0922 - 0.995i)T \)
good2 \( 1 + (-0.850 + 0.526i)T^{2} \)
5 \( 1 + (-0.932 + 0.361i)T^{2} \)
7 \( 1 + (0.890 - 0.811i)T + (0.0922 - 0.995i)T^{2} \)
11 \( 1 + (0.850 - 0.526i)T^{2} \)
13 \( 1 + (-1.25 + 1.14i)T + (0.0922 - 0.995i)T^{2} \)
17 \( 1 + (0.739 - 0.673i)T^{2} \)
19 \( 1 + (-1.18 - 1.56i)T + (-0.273 + 0.961i)T^{2} \)
23 \( 1 + (-0.850 - 0.526i)T^{2} \)
29 \( 1 + (0.932 - 0.361i)T^{2} \)
31 \( 1 + (1.78 + 0.887i)T + (0.602 + 0.798i)T^{2} \)
37 \( 1 + (1.34 + 0.124i)T + (0.982 + 0.183i)T^{2} \)
41 \( 1 + (0.932 + 0.361i)T^{2} \)
43 \( 1 + (0.719 - 0.0666i)T + (0.982 - 0.183i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.273 - 0.961i)T^{2} \)
59 \( 1 + (0.0922 + 0.995i)T^{2} \)
61 \( 1 + (0.831 - 0.322i)T + (0.739 - 0.673i)T^{2} \)
67 \( 1 + (1.29 - 1.42i)T + (-0.0922 - 0.995i)T^{2} \)
71 \( 1 + (-0.932 - 0.361i)T^{2} \)
73 \( 1 + (0.328 - 1.75i)T + (-0.932 - 0.361i)T^{2} \)
79 \( 1 + (-1.45 + 0.271i)T + (0.932 - 0.361i)T^{2} \)
83 \( 1 + (0.0922 - 0.995i)T^{2} \)
89 \( 1 + (-0.445 - 0.895i)T^{2} \)
97 \( 1 + (1.73 + 0.673i)T + (0.739 + 0.673i)T^{2} \)
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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

−10.30334797414372138009552866985, −9.543019296265112743861483966664, −8.605891120015539001297431690569, −7.66822633977436135921483151097, −6.71719397281816400152286951371, −5.69330653497104818960224248954, −5.59033065215566356151963646474, −3.61110946381950929149995450519, −2.89018719461219868752991161707, −1.47957576364833604445925150765, 1.58081593725565881571705110467, 3.15952886210902868900266499707, 3.67090671150344828342650835377, 5.01625382553620879097753442346, 6.37601724764579407991289361807, 6.90256089443383948312416448355, 7.48213928663341507530607196834, 8.802371219347208201961045264223, 9.299711899541565683435274175745, 10.58995522484664103468629283576

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