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Properties

Label 2-927-103.39-c0-0-0
Degree $2$
Conductor $927$
Sign $0.0352 + 0.999i$
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  + (−0.739 + 0.673i)4-s + (−0.876 − 1.75i)7-s + (−0.658 − 1.32i)13-s + (0.0922 − 0.995i)16-s + (0.538 + 0.100i)19-s + (−0.850 − 0.526i)25-s + (1.83 + 0.710i)28-s + (1.58 + 0.147i)31-s + (−1.42 − 1.07i)37-s + (0.840 − 0.634i)43-s + (−1.72 + 2.28i)49-s + (1.37 + 0.533i)52-s + (0.156 + 0.0971i)61-s + (0.602 + 0.798i)64-s + (0.646 + 0.322i)67-s + ⋯
L(s)  = 1  + (−0.739 + 0.673i)4-s + (−0.876 − 1.75i)7-s + (−0.658 − 1.32i)13-s + (0.0922 − 0.995i)16-s + (0.538 + 0.100i)19-s + (−0.850 − 0.526i)25-s + (1.83 + 0.710i)28-s + (1.58 + 0.147i)31-s + (−1.42 − 1.07i)37-s + (0.840 − 0.634i)43-s + (−1.72 + 2.28i)49-s + (1.37 + 0.533i)52-s + (0.156 + 0.0971i)61-s + (0.602 + 0.798i)64-s + (0.646 + 0.322i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0352 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0352 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.0352 + 0.999i$
Analytic conductor: \(0.462633\)
Root analytic conductor: \(0.680171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 927,\ (\ :0),\ 0.0352 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6000876932\)
\(L(\frac12)\) \(\approx\) \(0.6000876932\)
\(L(1)\) 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 \( 1 + (-0.602 + 0.798i)T \)
good2 \( 1 + (0.739 - 0.673i)T^{2} \)
5 \( 1 + (0.850 + 0.526i)T^{2} \)
7 \( 1 + (0.876 + 1.75i)T + (-0.602 + 0.798i)T^{2} \)
11 \( 1 + (-0.739 + 0.673i)T^{2} \)
13 \( 1 + (0.658 + 1.32i)T + (-0.602 + 0.798i)T^{2} \)
17 \( 1 + (0.445 + 0.895i)T^{2} \)
19 \( 1 + (-0.538 - 0.100i)T + (0.932 + 0.361i)T^{2} \)
23 \( 1 + (0.739 + 0.673i)T^{2} \)
29 \( 1 + (-0.850 - 0.526i)T^{2} \)
31 \( 1 + (-1.58 - 0.147i)T + (0.982 + 0.183i)T^{2} \)
37 \( 1 + (1.42 + 1.07i)T + (0.273 + 0.961i)T^{2} \)
41 \( 1 + (-0.850 + 0.526i)T^{2} \)
43 \( 1 + (-0.840 + 0.634i)T + (0.273 - 0.961i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.932 - 0.361i)T^{2} \)
59 \( 1 + (-0.602 - 0.798i)T^{2} \)
61 \( 1 + (-0.156 - 0.0971i)T + (0.445 + 0.895i)T^{2} \)
67 \( 1 + (-0.646 - 0.322i)T + (0.602 + 0.798i)T^{2} \)
71 \( 1 + (0.850 - 0.526i)T^{2} \)
73 \( 1 + (1.91 - 0.544i)T + (0.850 - 0.526i)T^{2} \)
79 \( 1 + (-0.243 + 0.857i)T + (-0.850 - 0.526i)T^{2} \)
83 \( 1 + (-0.602 + 0.798i)T^{2} \)
89 \( 1 + (-0.0922 - 0.995i)T^{2} \)
97 \( 1 + (1.44 - 0.895i)T + (0.445 - 0.895i)T^{2} \)
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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

−10.14641531654957792443784377512, −9.386415382998491614824703051760, −8.233295858507866131872550582159, −7.55274141226645013388896258472, −6.94222306650375960612071806703, −5.66063485894015826801718534132, −4.53346385568254425999423209211, −3.74152827259960863041238808559, −2.93357805405892855154879204913, −0.59317343365352584630477050730, 1.89509596668334793348985322019, 3.06635737434247060274363102942, 4.42952714737733080484791100739, 5.32327288493984282097771659875, 6.07225656869739471651709779407, 6.84697506459027074448682654740, 8.264841691405114164648321500916, 9.035229139678776916418773365935, 9.559763336800044900555310753880, 10.09820425030218644095440114486

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