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Properties

Label 1-1441-1441.503-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.246 - 0.969i$
Analytic cond. $6.69197$
Root an. cond. $6.69197$
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  + (0.836 + 0.548i)2-s + (−0.943 + 0.331i)3-s + (0.399 + 0.916i)4-s + (−0.607 − 0.794i)5-s + (−0.970 − 0.239i)6-s + (0.527 + 0.849i)7-s + (−0.168 + 0.985i)8-s + (0.779 − 0.626i)9-s + (−0.0724 − 0.997i)10-s + (−0.681 − 0.732i)12-s + (−0.926 − 0.377i)13-s + (−0.0241 + 0.999i)14-s + (0.836 + 0.548i)15-s + (−0.681 + 0.732i)16-s + (0.120 − 0.992i)17-s + (0.995 − 0.0965i)18-s + ⋯
L(s)  = 1  + (0.836 + 0.548i)2-s + (−0.943 + 0.331i)3-s + (0.399 + 0.916i)4-s + (−0.607 − 0.794i)5-s + (−0.970 − 0.239i)6-s + (0.527 + 0.849i)7-s + (−0.168 + 0.985i)8-s + (0.779 − 0.626i)9-s + (−0.0724 − 0.997i)10-s + (−0.681 − 0.732i)12-s + (−0.926 − 0.377i)13-s + (−0.0241 + 0.999i)14-s + (0.836 + 0.548i)15-s + (−0.681 + 0.732i)16-s + (0.120 − 0.992i)17-s + (0.995 − 0.0965i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.246 - 0.969i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (503, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.246 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4117252801 - 0.3199829297i\)
\(L(\frac12)\) \(\approx\) \(0.4117252801 - 0.3199829297i\)
\(L(1)\) \(\approx\) \(0.8871783102 + 0.2894206494i\)
\(L(1)\) \(\approx\) \(0.8871783102 + 0.2894206494i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.836 + 0.548i)T \)
3 \( 1 + (-0.943 + 0.331i)T \)
5 \( 1 + (-0.607 - 0.794i)T \)
7 \( 1 + (0.527 + 0.849i)T \)
13 \( 1 + (-0.926 - 0.377i)T \)
17 \( 1 + (0.120 - 0.992i)T \)
19 \( 1 + (-0.681 - 0.732i)T \)
23 \( 1 + (0.168 + 0.985i)T \)
29 \( 1 + (-0.998 - 0.0483i)T \)
31 \( 1 + (0.168 - 0.985i)T \)
37 \( 1 + (0.861 + 0.506i)T \)
41 \( 1 + (-0.120 - 0.992i)T \)
43 \( 1 + (-0.0241 - 0.999i)T \)
47 \( 1 + (-0.836 - 0.548i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.906 - 0.421i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.0241 + 0.999i)T \)
71 \( 1 + (0.168 - 0.985i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.607 - 0.794i)T \)
83 \( 1 + (-0.861 + 0.506i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.527 + 0.849i)T \)
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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

−21.26441266233565332449681946305, −19.971483953589052543457624080994, −19.507863064957490614468886145570, −18.70288838163617501257139952666, −18.08554523549420756904314971940, −16.9816437594758705694015936007, −16.47789757714673224721652813200, −15.39767082404320582970979544534, −14.491689764271402376509065446807, −14.293503801282000901924275339827, −12.89964408801437328656764693018, −12.58189866686945961377617193212, −11.566201602820992292273369501312, −11.08139110105467663739207370331, −10.45202646022302768901435243018, −9.84272720337905267622486109671, −8.10127521144306985354476045091, −7.33323710708290174835388954365, −6.58036486497395021217567970951, −5.96494624743110585395523273140, −4.66902694444450486744490688349, −4.36056540865189523264482799419, −3.32591538189323839810035148685, −2.134535929851552310208342817966, −1.28366100941928925256654128405, 0.16610172905828013020388603244, 1.83853957876051339904047649971, 3.03567478153116479973031472904, 4.15471999745057884059887878048, 4.886638230257266724869583365739, 5.29704677375756504018174960220, 6.08624212901535656319726955259, 7.25737153318299086702804404592, 7.7742346153370043850777452923, 8.89995509015770955384868683546, 9.573098244150459009351059673504, 11.04109536264789294708063153764, 11.584358015765738814165854749113, 12.15301340952948659606774993927, 12.786104019317271373061905323176, 13.5977069857459403751590159370, 14.928562024069388328824919008159, 15.28805345775757231638920283695, 15.86142200417844906543168346210, 16.84910444717543617523576996398, 17.167515048300267383055251117985, 18.036794871152383760604848066945, 18.99585109388604856651385618941, 20.15124928494841225897332251391, 20.79584667916558776011307430935

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