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Properties

Label 1-33e2-1089.659-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.598 + 0.800i$
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  + (−0.888 − 0.458i)2-s + (0.580 + 0.814i)4-s + (0.786 + 0.618i)5-s + (−0.981 − 0.189i)7-s + (−0.142 − 0.989i)8-s + (−0.415 − 0.909i)10-s + (−0.580 − 0.814i)13-s + (0.786 + 0.618i)14-s + (−0.327 + 0.945i)16-s + (−0.959 − 0.281i)17-s + (0.959 − 0.281i)19-s + (−0.0475 + 0.998i)20-s + (−0.981 + 0.189i)23-s + (0.235 + 0.971i)25-s + (0.142 + 0.989i)26-s + ⋯
L(s)  = 1  + (−0.888 − 0.458i)2-s + (0.580 + 0.814i)4-s + (0.786 + 0.618i)5-s + (−0.981 − 0.189i)7-s + (−0.142 − 0.989i)8-s + (−0.415 − 0.909i)10-s + (−0.580 − 0.814i)13-s + (0.786 + 0.618i)14-s + (−0.327 + 0.945i)16-s + (−0.959 − 0.281i)17-s + (0.959 − 0.281i)19-s + (−0.0475 + 0.998i)20-s + (−0.981 + 0.189i)23-s + (0.235 + 0.971i)25-s + (0.142 + 0.989i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.598 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.598 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.598 + 0.800i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.598 + 0.800i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6685792604 + 0.3349684394i\)
\(L(\frac12)\) \(\approx\) \(0.6685792604 + 0.3349684394i\)
\(L(1)\) \(\approx\) \(0.6847594403 + 0.001273054402i\)
\(L(1)\) \(\approx\) \(0.6847594403 + 0.001273054402i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.888 - 0.458i)T \)
5 \( 1 + (0.786 + 0.618i)T \)
7 \( 1 + (-0.981 - 0.189i)T \)
13 \( 1 + (-0.580 - 0.814i)T \)
17 \( 1 + (-0.959 - 0.281i)T \)
19 \( 1 + (0.959 - 0.281i)T \)
23 \( 1 + (-0.981 + 0.189i)T \)
29 \( 1 + (0.723 + 0.690i)T \)
31 \( 1 + (-0.995 - 0.0950i)T \)
37 \( 1 + (0.415 + 0.909i)T \)
41 \( 1 + (0.0475 + 0.998i)T \)
43 \( 1 + (0.786 - 0.618i)T \)
47 \( 1 + (-0.0475 + 0.998i)T \)
53 \( 1 + (0.654 - 0.755i)T \)
59 \( 1 + (0.888 - 0.458i)T \)
61 \( 1 + (-0.0475 + 0.998i)T \)
67 \( 1 + (0.0475 + 0.998i)T \)
71 \( 1 + (0.959 - 0.281i)T \)
73 \( 1 + (0.654 + 0.755i)T \)
79 \( 1 + (-0.928 + 0.371i)T \)
83 \( 1 + (0.981 + 0.189i)T \)
89 \( 1 + (0.959 + 0.281i)T \)
97 \( 1 + (-0.786 + 0.618i)T \)
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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

−21.30894438728261773555429137103, −20.17085547935383796902538714013, −19.81573083643499419867067443243, −18.91490180893187932436234328464, −18.07903337141077225041785891761, −17.47535543005881122652985193235, −16.53553487140950575386494947277, −16.19114347870821671807288149258, −15.346294503374277416798481196969, −14.22924078399356433546072876000, −13.65815912678867676182830696410, −12.56287031116204964660335465629, −11.84525074127868919701755425470, −10.67863300909055921477575186765, −9.83797274643160924873410340905, −9.324269765912465456875937360178, −8.70618059432972641091465511127, −7.59855688394958704321129432017, −6.67387027043365349167455331538, −6.01762351798087888611817140390, −5.229225709582505488115975569566, −4.05167522845455596726202097270, −2.49520580023004720356763665774, −1.86503621153863833018244923937, −0.4707550884678239831702811862, 1.01485669390690728278636256797, 2.36254410036080992360149331916, 2.895391685373550923139208457765, 3.83834395160028442068917994578, 5.33043457370039073014724580309, 6.39619618460608113767226305177, 7.00949908230162879934289729741, 7.843404189031201115869710887938, 9.02715199523572695309792008636, 9.70379972035031851332768564374, 10.21838767472094881290483613556, 11.011817165440576153369306957159, 11.934015763198064473902443526430, 12.91319135823812877375027363855, 13.4347798100117397771037498933, 14.501518767067569023231827428227, 15.59419325568807527834519670164, 16.17983843329014344028257475301, 17.1098766082818024545717082761, 17.90310026815390852352984524372, 18.24607746401545859834605223200, 19.27684657017247994854403825838, 19.97452351695612832262629068172, 20.45577829187398694653314240252, 21.66846301290152913524507914932

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