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Properties

Label 1-1441-1441.97-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.769 - 0.638i$
Analytic cond. $154.856$
Root an. cond. $154.856$
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  + (−0.926 + 0.377i)2-s + (0.485 + 0.873i)3-s + (0.715 − 0.698i)4-s + (0.981 − 0.192i)5-s + (−0.779 − 0.626i)6-s + (−0.354 − 0.935i)7-s + (−0.399 + 0.916i)8-s + (−0.527 + 0.849i)9-s + (−0.836 + 0.548i)10-s + (0.958 + 0.285i)12-s + (−0.354 − 0.935i)13-s + (0.681 + 0.732i)14-s + (0.644 + 0.764i)15-s + (0.0241 − 0.999i)16-s + (−0.958 + 0.285i)17-s + (0.168 − 0.985i)18-s + ⋯
L(s)  = 1  + (−0.926 + 0.377i)2-s + (0.485 + 0.873i)3-s + (0.715 − 0.698i)4-s + (0.981 − 0.192i)5-s + (−0.779 − 0.626i)6-s + (−0.354 − 0.935i)7-s + (−0.399 + 0.916i)8-s + (−0.527 + 0.849i)9-s + (−0.836 + 0.548i)10-s + (0.958 + 0.285i)12-s + (−0.354 − 0.935i)13-s + (0.681 + 0.732i)14-s + (0.644 + 0.764i)15-s + (0.0241 − 0.999i)16-s + (−0.958 + 0.285i)17-s + (0.168 − 0.985i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.769 - 0.638i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.769 - 0.638i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.280045321 - 0.4615349278i\)
\(L(\frac12)\) \(\approx\) \(1.280045321 - 0.4615349278i\)
\(L(1)\) \(\approx\) \(0.8674231149 + 0.1767382814i\)
\(L(1)\) \(\approx\) \(0.8674231149 + 0.1767382814i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.926 + 0.377i)T \)
3 \( 1 + (0.485 + 0.873i)T \)
5 \( 1 + (0.981 - 0.192i)T \)
7 \( 1 + (-0.354 - 0.935i)T \)
13 \( 1 + (-0.354 - 0.935i)T \)
17 \( 1 + (-0.958 + 0.285i)T \)
19 \( 1 + (0.607 + 0.794i)T \)
23 \( 1 + (-0.215 + 0.976i)T \)
29 \( 1 + (0.970 - 0.239i)T \)
31 \( 1 + (0.861 - 0.506i)T \)
37 \( 1 + (-0.885 - 0.464i)T \)
41 \( 1 + (-0.943 + 0.331i)T \)
43 \( 1 + (0.981 - 0.192i)T \)
47 \( 1 + (0.527 - 0.849i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.958 - 0.285i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (-0.981 - 0.192i)T \)
71 \( 1 + (-0.995 - 0.0965i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (0.681 - 0.732i)T \)
83 \( 1 + (-0.715 - 0.698i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.998 - 0.0483i)T \)
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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

−20.58830880695533475500814730331, −19.52483474236652806926061467977, −19.15047095002909331429546703601, −18.33080086052757128835986374505, −17.82817027379407698148939527921, −17.27555158804692417539775289537, −16.18760779217505896754392078176, −15.45312736856194049733928765508, −14.419886102451862506626100113348, −13.66751180320587110275554493259, −12.89776607261806704865747114160, −12.107288509551127032503395100847, −11.56335853522560495610264322357, −10.42865008349927747787732838367, −9.58670089723526488459025776562, −8.86684321198753740879706346762, −8.58513084663992318349215738609, −7.209190495901853224525909477, −6.66682501144562509219927806834, −6.07669474723279022380210004189, −4.71524027924458919342541788085, −3.11225792129224206071071861944, −2.52735392621346544442789240690, −1.96054925362391117124030784692, −0.93066986969333553591142766972, 0.35086292079867299329753731776, 1.52580201319085042491417449209, 2.51573282227779420895678339295, 3.4410504801724212261624963119, 4.65338012567612627433310891663, 5.516913567384647224303390660789, 6.2786950550666633592155522209, 7.32953122474051161683800593574, 8.10565174653721923291069870983, 8.9222709415742181584154392188, 9.71897018264032959859823010557, 10.232146952403249554100124260424, 10.63678225521887157778663305721, 11.77541068777446432907387463956, 13.09855086073627608411224478125, 13.860015458767330446877176946872, 14.4000425121874634992268543918, 15.48097376474581430651515977466, 15.88366278903752859999924185565, 16.85251539671400560204879978288, 17.34290644881760349376968151152, 17.88544562366244869017411386890, 19.10207065011871009557754044476, 19.76074572520411251267517297181, 20.445896431764108887798899948176

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