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Properties

Label 1-224-224.69-r1-0-0
Degree $1$
Conductor $224$
Sign $-0.831 + 0.555i$
Analytic cond. $24.0721$
Root an. cond. $24.0721$
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  + (0.707 − 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

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

Invariants

Degree: \(1\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.831 + 0.555i$
Analytic conductor: \(24.0721\)
Root analytic conductor: \(24.0721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 224,\ (1:\ ),\ -0.831 + 0.555i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1488432530 - 0.4906704478i\)
\(L(\frac12)\) \(\approx\) \(-0.1488432530 - 0.4906704478i\)
\(L(1)\) \(\approx\) \(0.8087848797 - 0.3925295947i\)
\(L(1)\) \(\approx\) \(0.8087848797 - 0.3925295947i\)

Euler product

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

−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 $Z$-function along the critical line