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Properties

Label 1-33e2-1089.115-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.946 - 0.321i$
Analytic cond. $5.05729$
Root an. cond. $5.05729$
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  + (0.988 − 0.151i)2-s + (0.953 − 0.299i)4-s + (0.272 + 0.962i)5-s + (0.483 − 0.875i)7-s + (0.897 − 0.441i)8-s + (0.415 + 0.909i)10-s + (−0.948 − 0.318i)13-s + (0.345 − 0.938i)14-s + (0.820 − 0.572i)16-s + (0.610 + 0.791i)17-s + (−0.564 − 0.825i)19-s + (0.548 + 0.836i)20-s + (0.981 − 0.189i)23-s + (−0.851 + 0.524i)25-s + (−0.985 − 0.170i)26-s + ⋯
L(s)  = 1  + (0.988 − 0.151i)2-s + (0.953 − 0.299i)4-s + (0.272 + 0.962i)5-s + (0.483 − 0.875i)7-s + (0.897 − 0.441i)8-s + (0.415 + 0.909i)10-s + (−0.948 − 0.318i)13-s + (0.345 − 0.938i)14-s + (0.820 − 0.572i)16-s + (0.610 + 0.791i)17-s + (−0.564 − 0.825i)19-s + (0.548 + 0.836i)20-s + (0.981 − 0.189i)23-s + (−0.851 + 0.524i)25-s + (−0.985 − 0.170i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.946 - 0.321i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.946 - 0.321i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.295494148 - 0.5448818141i\)
\(L(\frac12)\) \(\approx\) \(3.295494148 - 0.5448818141i\)
\(L(1)\) \(\approx\) \(2.144852278 - 0.1927262381i\)
\(L(1)\) \(\approx\) \(2.144852278 - 0.1927262381i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.988 - 0.151i)T \)
5 \( 1 + (0.272 + 0.962i)T \)
7 \( 1 + (0.483 - 0.875i)T \)
13 \( 1 + (-0.948 - 0.318i)T \)
17 \( 1 + (0.610 + 0.791i)T \)
19 \( 1 + (-0.564 - 0.825i)T \)
23 \( 1 + (0.981 - 0.189i)T \)
29 \( 1 + (0.879 - 0.475i)T \)
31 \( 1 + (0.861 - 0.508i)T \)
37 \( 1 + (0.993 - 0.113i)T \)
41 \( 1 + (-0.935 + 0.353i)T \)
43 \( 1 + (-0.786 + 0.618i)T \)
47 \( 1 + (0.964 - 0.263i)T \)
53 \( 1 + (0.0855 - 0.996i)T \)
59 \( 1 + (0.161 + 0.986i)T \)
61 \( 1 + (-0.625 + 0.780i)T \)
67 \( 1 + (0.0475 + 0.998i)T \)
71 \( 1 + (0.941 + 0.336i)T \)
73 \( 1 + (-0.921 + 0.389i)T \)
79 \( 1 + (-0.532 + 0.846i)T \)
83 \( 1 + (-0.683 - 0.730i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (0.272 - 0.962i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.51175282701043707813290524666, −20.91229475096652156133150071926, −20.22485377200379098574844307196, −19.31476854482500479317940032157, −18.42315447560237832963380974645, −17.185293472629273387956518924215, −16.83417955198680956688570246634, −15.86111116900021154655395243157, −15.18590041009558451789688559158, −14.32578438260219390892928599906, −13.73170406598046212563899341601, −12.61793411403123991168980790353, −12.211910614387527621785726717612, −11.60283341899760432651042206054, −10.422482704190713063593068593578, −9.43241702804085800718161141000, −8.501964947118068439921014921001, −7.73935656899243087197719960518, −6.67913087104959111549130914978, −5.71044660834011835630163087190, −4.99093482067853513510628859387, −4.53412842143452044666986864815, −3.16407280790188658704796209514, −2.260751426096055022714545910788, −1.33262465219549539610365091486, 1.13270549427785591856673702350, 2.37288000283945073191401401149, 3.02318787411973625027483900271, 4.11407472469929556284887811525, 4.82972931690927563069504469717, 5.869712176389909954212746675485, 6.76834463461385776159345913950, 7.32811715332351934353653458408, 8.26171818934841081891133276940, 9.96290572617696871479918813286, 10.30336524421014063228641036686, 11.18496671236514842183218116155, 11.82177928412309529229872078544, 12.98339547746274172352012035799, 13.49946289967823463456164488723, 14.48920359200717404034625247255, 14.82574423317217963050002507603, 15.56586785094860881176613032686, 16.9538648844158148039212294224, 17.20026220110888885612620231001, 18.38150064286772586968361801887, 19.42514733348374796436327731509, 19.77060080865107889329384597510, 20.90747732698354445924804417042, 21.45826942449262864120166853808

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