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Properties

Label 2-921-921.716-c0-0-0
Degree $2$
Conductor $921$
Sign $0.999 + 0.0416i$
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  + (−0.982 − 0.183i)3-s + (0.739 − 0.673i)4-s + (0.726 + 0.961i)7-s + (0.932 + 0.361i)9-s + (−0.850 + 0.526i)12-s + (−0.156 + 1.69i)13-s + (0.0922 − 0.995i)16-s + (0.0822 − 0.887i)19-s + (−0.537 − 1.07i)21-s + (0.0922 + 0.995i)25-s + (−0.850 − 0.526i)27-s + (1.18 + 0.221i)28-s + (1.18 − 0.221i)31-s + (0.932 − 0.361i)36-s + (−0.890 − 1.17i)37-s + ⋯
L(s)  = 1  + (−0.982 − 0.183i)3-s + (0.739 − 0.673i)4-s + (0.726 + 0.961i)7-s + (0.932 + 0.361i)9-s + (−0.850 + 0.526i)12-s + (−0.156 + 1.69i)13-s + (0.0922 − 0.995i)16-s + (0.0822 − 0.887i)19-s + (−0.537 − 1.07i)21-s + (0.0922 + 0.995i)25-s + (−0.850 − 0.526i)27-s + (1.18 + 0.221i)28-s + (1.18 − 0.221i)31-s + (0.932 − 0.361i)36-s + (−0.890 − 1.17i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(921\)    =    \(3 \cdot 307\)
Sign: $0.999 + 0.0416i$
Analytic conductor: \(0.459638\)
Root analytic conductor: \(0.677966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{921} (716, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 921,\ (\ :0),\ 0.999 + 0.0416i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9548023483\)
\(L(\frac12)\) \(\approx\) \(0.9548023483\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.982 + 0.183i)T \)
307 \( 1 + (0.982 + 0.183i)T \)
good2 \( 1 + (-0.739 + 0.673i)T^{2} \)
5 \( 1 + (-0.0922 - 0.995i)T^{2} \)
7 \( 1 + (-0.726 - 0.961i)T + (-0.273 + 0.961i)T^{2} \)
11 \( 1 + (-0.932 + 0.361i)T^{2} \)
13 \( 1 + (0.156 - 1.69i)T + (-0.982 - 0.183i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-0.0822 + 0.887i)T + (-0.982 - 0.183i)T^{2} \)
23 \( 1 + (-0.739 - 0.673i)T^{2} \)
29 \( 1 + (-0.739 + 0.673i)T^{2} \)
31 \( 1 + (-1.18 + 0.221i)T + (0.932 - 0.361i)T^{2} \)
37 \( 1 + (0.890 + 1.17i)T + (-0.273 + 0.961i)T^{2} \)
41 \( 1 + (0.273 + 0.961i)T^{2} \)
43 \( 1 + (-0.658 + 1.32i)T + (-0.602 - 0.798i)T^{2} \)
47 \( 1 + (0.982 + 0.183i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.850 + 0.526i)T^{2} \)
61 \( 1 + (0.757 - 1.52i)T + (-0.602 - 0.798i)T^{2} \)
67 \( 1 + (0.156 - 0.0971i)T + (0.445 - 0.895i)T^{2} \)
71 \( 1 + (0.602 - 0.798i)T^{2} \)
73 \( 1 + (0.111 - 1.20i)T + (-0.982 - 0.183i)T^{2} \)
79 \( 1 + (0.876 + 1.75i)T + (-0.602 + 0.798i)T^{2} \)
83 \( 1 + (0.273 + 0.961i)T^{2} \)
89 \( 1 + (-0.445 + 0.895i)T^{2} \)
97 \( 1 + (0.156 + 1.69i)T + (-0.982 + 0.183i)T^{2} \)
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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

−10.53245902790612926836496041290, −9.487211358680401561370602826459, −8.780735646195658939901437358592, −7.35127779709497804075590582113, −6.85040071545582000128686776762, −5.88587241992242535841188741913, −5.23657301916201584593493726321, −4.35205353782291509648129236075, −2.40965089399203839295174555156, −1.53539023738219650093148732240, 1.28944719197253871773365869213, 2.98751484942557883898586593779, 4.09012777844073195974884420129, 5.00431202476316081583477469039, 6.06773058242154867436281446014, 6.83873000141475779385267629579, 7.87837737386554676740144303393, 8.127467367275148265793143588970, 9.854806813514014258972266984254, 10.55642184063010481676842394163

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