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Properties

Label 2-921-921.830-c0-0-0
Degree $2$
Conductor $921$
Sign $-0.0543 + 0.998i$
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  + (−0.850 − 0.526i)3-s + (−0.602 − 0.798i)4-s + (1.73 − 0.673i)7-s + (0.445 + 0.895i)9-s + (0.0922 + 0.995i)12-s + (−0.0505 + 0.177i)13-s + (−0.273 + 0.961i)16-s + (0.538 − 1.89i)19-s + (−1.83 − 0.342i)21-s + (−0.273 − 0.961i)25-s + (0.0922 − 0.995i)27-s + (−1.58 − 0.981i)28-s + (−1.58 + 0.981i)31-s + (0.445 − 0.895i)36-s + (−1.12 + 0.435i)37-s + ⋯
L(s)  = 1  + (−0.850 − 0.526i)3-s + (−0.602 − 0.798i)4-s + (1.73 − 0.673i)7-s + (0.445 + 0.895i)9-s + (0.0922 + 0.995i)12-s + (−0.0505 + 0.177i)13-s + (−0.273 + 0.961i)16-s + (0.538 − 1.89i)19-s + (−1.83 − 0.342i)21-s + (−0.273 − 0.961i)25-s + (0.0922 − 0.995i)27-s + (−1.58 − 0.981i)28-s + (−1.58 + 0.981i)31-s + (0.445 − 0.895i)36-s + (−1.12 + 0.435i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0543 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0543 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(921\)    =    \(3 \cdot 307\)
Sign: $-0.0543 + 0.998i$
Analytic conductor: \(0.459638\)
Root analytic conductor: \(0.677966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{921} (830, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 921,\ (\ :0),\ -0.0543 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7731296763\)
\(L(\frac12)\) \(\approx\) \(0.7731296763\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.850 + 0.526i)T \)
307 \( 1 + (0.850 + 0.526i)T \)
good2 \( 1 + (0.602 + 0.798i)T^{2} \)
5 \( 1 + (0.273 + 0.961i)T^{2} \)
7 \( 1 + (-1.73 + 0.673i)T + (0.739 - 0.673i)T^{2} \)
11 \( 1 + (-0.445 + 0.895i)T^{2} \)
13 \( 1 + (0.0505 - 0.177i)T + (-0.850 - 0.526i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-0.538 + 1.89i)T + (-0.850 - 0.526i)T^{2} \)
23 \( 1 + (0.602 - 0.798i)T^{2} \)
29 \( 1 + (0.602 + 0.798i)T^{2} \)
31 \( 1 + (1.58 - 0.981i)T + (0.445 - 0.895i)T^{2} \)
37 \( 1 + (1.12 - 0.435i)T + (0.739 - 0.673i)T^{2} \)
41 \( 1 + (-0.739 - 0.673i)T^{2} \)
43 \( 1 + (-1.18 + 0.221i)T + (0.932 - 0.361i)T^{2} \)
47 \( 1 + (0.850 + 0.526i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.0922 + 0.995i)T^{2} \)
61 \( 1 + (0.181 - 0.0339i)T + (0.932 - 0.361i)T^{2} \)
67 \( 1 + (0.0505 + 0.544i)T + (-0.982 + 0.183i)T^{2} \)
71 \( 1 + (-0.932 - 0.361i)T^{2} \)
73 \( 1 + (0.510 - 1.79i)T + (-0.850 - 0.526i)T^{2} \)
79 \( 1 + (-1.67 - 0.312i)T + (0.932 + 0.361i)T^{2} \)
83 \( 1 + (-0.739 - 0.673i)T^{2} \)
89 \( 1 + (0.982 - 0.183i)T^{2} \)
97 \( 1 + (0.0505 + 0.177i)T + (-0.850 + 0.526i)T^{2} \)
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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

−10.40478101313746172993432026226, −9.211967071108291544074051494227, −8.359985557979706181269487745064, −7.40095269742173294670560665025, −6.71470601242447009402544512772, −5.42825815751879305928600267490, −4.97047050655074953618943576609, −4.21304321135955256272398661921, −2.02144005581770740486702236152, −0.962420778966855683391805750258, 1.74142456870381037381032777817, 3.52761002484302052406241891388, 4.34094631759496718511486728379, 5.32665885942514729356324038828, 5.73208579139219437476047542569, 7.41148267210522333797944791459, 7.925238541432354553442792391755, 8.897723530692246772159655417601, 9.554765557853472896206266912661, 10.68649132841930094556364648810

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