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Properties

Label 2-921-921.728-c0-0-0
Degree $2$
Conductor $921$
Sign $0.177 + 0.984i$
Analytic cond. $0.459638$
Root an. cond. $0.677966$
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)3-s + (0.932 − 0.361i)4-s + (0.397 − 0.798i)7-s + (−0.982 − 0.183i)9-s + (−0.273 − 0.961i)12-s + (−0.404 + 0.368i)13-s + (0.739 − 0.673i)16-s + (−1.25 + 1.14i)19-s + (−0.757 − 0.469i)21-s + (0.739 + 0.673i)25-s + (−0.273 + 0.961i)27-s + (0.0822 − 0.887i)28-s + (0.0822 + 0.887i)31-s + (−0.982 + 0.183i)36-s + (0.831 − 1.66i)37-s + ⋯
L(s)  = 1  + (0.0922 − 0.995i)3-s + (0.932 − 0.361i)4-s + (0.397 − 0.798i)7-s + (−0.982 − 0.183i)9-s + (−0.273 − 0.961i)12-s + (−0.404 + 0.368i)13-s + (0.739 − 0.673i)16-s + (−1.25 + 1.14i)19-s + (−0.757 − 0.469i)21-s + (0.739 + 0.673i)25-s + (−0.273 + 0.961i)27-s + (0.0822 − 0.887i)28-s + (0.0822 + 0.887i)31-s + (−0.982 + 0.183i)36-s + (0.831 − 1.66i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.177 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.177 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(921\)    =    \(3 \cdot 307\)
Sign: $0.177 + 0.984i$
Analytic conductor: \(0.459638\)
Root analytic conductor: \(0.677966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{921} (728, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 921,\ (\ :0),\ 0.177 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.235091026\)
\(L(\frac12)\) \(\approx\) \(1.235091026\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0922 + 0.995i)T \)
307 \( 1 + (-0.0922 + 0.995i)T \)
good2 \( 1 + (-0.932 + 0.361i)T^{2} \)
5 \( 1 + (-0.739 - 0.673i)T^{2} \)
7 \( 1 + (-0.397 + 0.798i)T + (-0.602 - 0.798i)T^{2} \)
11 \( 1 + (0.982 - 0.183i)T^{2} \)
13 \( 1 + (0.404 - 0.368i)T + (0.0922 - 0.995i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (1.25 - 1.14i)T + (0.0922 - 0.995i)T^{2} \)
23 \( 1 + (-0.932 - 0.361i)T^{2} \)
29 \( 1 + (-0.932 + 0.361i)T^{2} \)
31 \( 1 + (-0.0822 - 0.887i)T + (-0.982 + 0.183i)T^{2} \)
37 \( 1 + (-0.831 + 1.66i)T + (-0.602 - 0.798i)T^{2} \)
41 \( 1 + (0.602 - 0.798i)T^{2} \)
43 \( 1 + (1.58 - 0.981i)T + (0.445 - 0.895i)T^{2} \)
47 \( 1 + (-0.0922 + 0.995i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.273 - 0.961i)T^{2} \)
61 \( 1 + (-0.465 + 0.288i)T + (0.445 - 0.895i)T^{2} \)
67 \( 1 + (0.404 + 1.42i)T + (-0.850 + 0.526i)T^{2} \)
71 \( 1 + (-0.445 - 0.895i)T^{2} \)
73 \( 1 + (-0.658 + 0.600i)T + (0.0922 - 0.995i)T^{2} \)
79 \( 1 + (0.156 + 0.0971i)T + (0.445 + 0.895i)T^{2} \)
83 \( 1 + (0.602 - 0.798i)T^{2} \)
89 \( 1 + (0.850 - 0.526i)T^{2} \)
97 \( 1 + (0.404 + 0.368i)T + (0.0922 + 0.995i)T^{2} \)
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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

−10.37832248646549592366690838498, −9.209189950419170844598470843878, −8.112073014624085454674608943259, −7.49717575249396683658626194678, −6.72626553826092588548374366915, −6.08695068245160772776051916220, −4.95272707704674969589009802115, −3.54967856420566166929930059918, −2.27404520036145013921286146386, −1.35833926087354270853112896696, 2.29165457711634037910696193500, 2.95461774199377805378793425274, 4.25990634018578108320860716651, 5.16027179520275002412170722927, 6.13126534208459290446862788177, 7.00748686087160736688998796048, 8.327824183586344374694742554311, 8.556544564447230151333051455934, 9.777085656078424114190013807115, 10.48193138806353299220862442005

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