Properties

Label 1-224-224.69-r1-0-0
Degree 11
Conductor 224224
Sign 0.831+0.555i-0.831 + 0.555i
Analytic cond. 24.072124.0721
Root an. cond. 24.072124.0721
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)3-s + (−0.707 − 0.707i)5-s i·9-s + (−0.707 − 0.707i)11-s + (−0.707 + 0.707i)13-s − 15-s + 17-s + (−0.707 + 0.707i)19-s i·23-s + i·25-s + (−0.707 − 0.707i)27-s + (−0.707 + 0.707i)29-s − 31-s − 33-s + (0.707 + 0.707i)37-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (−0.707 − 0.707i)5-s i·9-s + (−0.707 − 0.707i)11-s + (−0.707 + 0.707i)13-s − 15-s + 17-s + (−0.707 + 0.707i)19-s i·23-s + i·25-s + (−0.707 − 0.707i)27-s + (−0.707 + 0.707i)29-s − 31-s − 33-s + (0.707 + 0.707i)37-s + ⋯

Functional equation

Λ(s)=(224s/2ΓR(s+1)L(s)=((0.831+0.555i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(224s/2ΓR(s+1)L(s)=((0.831+0.555i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 224224    =    2572^{5} \cdot 7
Sign: 0.831+0.555i-0.831 + 0.555i
Analytic conductor: 24.072124.0721
Root analytic conductor: 24.072124.0721
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ224(69,)\chi_{224} (69, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 224, (1: ), 0.831+0.555i)(1,\ 224,\ (1:\ ),\ -0.831 + 0.555i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.14884325300.4906704478i-0.1488432530 - 0.4906704478i
L(12)L(\frac12) \approx 0.14884325300.4906704478i-0.1488432530 - 0.4906704478i
L(1)L(1) \approx 0.80878487970.3925295947i0.8087848797 - 0.3925295947i
L(1)L(1) \approx 0.80878487970.3925295947i0.8087848797 - 0.3925295947i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
7 1 1
good3 1+(0.7070.707i)T 1 + (0.707 - 0.707i)T
5 1+(0.7070.707i)T 1 + (-0.707 - 0.707i)T
11 1+(0.7070.707i)T 1 + (-0.707 - 0.707i)T
13 1+(0.707+0.707i)T 1 + (-0.707 + 0.707i)T
17 1+T 1 + T
19 1+(0.707+0.707i)T 1 + (-0.707 + 0.707i)T
23 1iT 1 - iT
29 1+(0.707+0.707i)T 1 + (-0.707 + 0.707i)T
31 1T 1 - T
37 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
41 1iT 1 - iT
43 1+(0.7070.707i)T 1 + (-0.707 - 0.707i)T
47 1+T 1 + T
53 1+(0.7070.707i)T 1 + (-0.707 - 0.707i)T
59 1+(0.7070.707i)T 1 + (-0.707 - 0.707i)T
61 1+(0.7070.707i)T 1 + (0.707 - 0.707i)T
67 1+(0.707+0.707i)T 1 + (-0.707 + 0.707i)T
71 1+iT 1 + iT
73 1iT 1 - iT
79 1T 1 - T
83 1+(0.707+0.707i)T 1 + (-0.707 + 0.707i)T
89 1iT 1 - iT
97 1T 1 - T
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   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−26.7807410997792503248388701980, −25.86021199040421333116246550308, −25.29522213939535190505896964201, −23.87049331786205293643819194932, −22.97550883057863650611067480531, −22.07536429993297743705638404541, −21.16790043681929923086174207049, −20.12949307904524493313936579676, −19.446002035807545827309932578801, −18.49113610573175001130197503582, −17.28152404487077069365684820346, −16.04959991253900158815303435083, −15.11845162489845469003467394663, −14.77874986159120159722000538506, −13.460157869240360618985968891988, −12.32844349916371484640214476642, −10.9998026853409395593396814360, −10.235958877994539236990185376836, −9.257303217835056581969721674304, −7.79440993579247640516951340877, −7.41974616460463472731677292148, −5.53915907423940944031274044590, −4.36197447562445711101246368430, −3.268095652948249509889159846205, −2.303435761224473848897725512, 0.146857417349801324303111432949, 1.575647848744257594288853392, 2.997993522598569551210490106305, 4.15693763100525821668276699175, 5.550054375572188654038012230910, 6.97021037578420985826933760833, 7.98650236676089701378589195059, 8.62218862458046360906599014343, 9.802663514685311181952366608676, 11.30155686321735178873750444697, 12.40153948013768588002467289021, 12.94453013671282807654851430402, 14.220904313684311676556008990782, 14.960174877112215911823854358873, 16.277186260206826504532000589912, 16.96176705138605275525090236238, 18.64603756829044773352300793557, 18.88728758116636040086568915673, 20.07518123687075188996521085329, 20.73510972755785830992688199182, 21.77065592980773533980924570367, 23.38354401449925596331320564939, 23.80146301386437925015039461105, 24.6650663757788501731086345899, 25.56900824399626183963736872624

Graph of the ZZ-function along the critical line