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Properties

Label 2-921-921.623-c0-0-0
Degree $2$
Conductor $921$
Sign $0.892 - 0.451i$
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  + (0.932 − 0.361i)3-s + (0.0922 + 0.995i)4-s + (0.149 + 0.526i)7-s + (0.739 − 0.673i)9-s + (0.445 + 0.895i)12-s + (−0.876 + 0.163i)13-s + (−0.982 + 0.183i)16-s + (1.18 − 0.221i)19-s + (0.329 + 0.436i)21-s + (−0.982 − 0.183i)25-s + (0.445 − 0.895i)27-s + (−0.510 + 0.197i)28-s + (−0.510 − 0.197i)31-s + (0.739 + 0.673i)36-s + (−0.0505 − 0.177i)37-s + ⋯
L(s)  = 1  + (0.932 − 0.361i)3-s + (0.0922 + 0.995i)4-s + (0.149 + 0.526i)7-s + (0.739 − 0.673i)9-s + (0.445 + 0.895i)12-s + (−0.876 + 0.163i)13-s + (−0.982 + 0.183i)16-s + (1.18 − 0.221i)19-s + (0.329 + 0.436i)21-s + (−0.982 − 0.183i)25-s + (0.445 − 0.895i)27-s + (−0.510 + 0.197i)28-s + (−0.510 − 0.197i)31-s + (0.739 + 0.673i)36-s + (−0.0505 − 0.177i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(921\)    =    \(3 \cdot 307\)
Sign: $0.892 - 0.451i$
Analytic conductor: \(0.459638\)
Root analytic conductor: \(0.677966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{921} (623, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 921,\ (\ :0),\ 0.892 - 0.451i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.363260539\)
\(L(\frac12)\) \(\approx\) \(1.363260539\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.932 + 0.361i)T \)
307 \( 1 + (-0.932 + 0.361i)T \)
good2 \( 1 + (-0.0922 - 0.995i)T^{2} \)
5 \( 1 + (0.982 + 0.183i)T^{2} \)
7 \( 1 + (-0.149 - 0.526i)T + (-0.850 + 0.526i)T^{2} \)
11 \( 1 + (-0.739 - 0.673i)T^{2} \)
13 \( 1 + (0.876 - 0.163i)T + (0.932 - 0.361i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-1.18 + 0.221i)T + (0.932 - 0.361i)T^{2} \)
23 \( 1 + (-0.0922 + 0.995i)T^{2} \)
29 \( 1 + (-0.0922 - 0.995i)T^{2} \)
31 \( 1 + (0.510 + 0.197i)T + (0.739 + 0.673i)T^{2} \)
37 \( 1 + (0.0505 + 0.177i)T + (-0.850 + 0.526i)T^{2} \)
41 \( 1 + (0.850 + 0.526i)T^{2} \)
43 \( 1 + (0.111 - 0.147i)T + (-0.273 - 0.961i)T^{2} \)
47 \( 1 + (-0.932 + 0.361i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.445 + 0.895i)T^{2} \)
61 \( 1 + (0.537 - 0.711i)T + (-0.273 - 0.961i)T^{2} \)
67 \( 1 + (0.876 + 1.75i)T + (-0.602 + 0.798i)T^{2} \)
71 \( 1 + (0.273 - 0.961i)T^{2} \)
73 \( 1 + (-0.538 + 0.100i)T + (0.932 - 0.361i)T^{2} \)
79 \( 1 + (1.12 + 1.48i)T + (-0.273 + 0.961i)T^{2} \)
83 \( 1 + (0.850 + 0.526i)T^{2} \)
89 \( 1 + (0.602 - 0.798i)T^{2} \)
97 \( 1 + (0.876 + 0.163i)T + (0.932 + 0.361i)T^{2} \)
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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

−10.09565193640313125916942742176, −9.253254137592886271393567215818, −8.702454892886657323157766645463, −7.59252984988674499020707454834, −7.46059072279389923654155631506, −6.24175683996477196860560072204, −4.90816050381542514208597283320, −3.80869787995154475423368802477, −2.89848139619868900052868654136, −2.00796755033165883610771443478, 1.51774645046580629128230582985, 2.71781111399605409835959880297, 3.93945251912481774679802262243, 4.90715390675142553892579076364, 5.72241297025731612021268219167, 7.07861074788866593488999948302, 7.60055056168493160851681552148, 8.682922238181329377646936656291, 9.668778822782667658984548084139, 9.941717685365357521878592787628

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