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Properties

Label 1-2176-2176.1971-r1-0-0
Degree $1$
Conductor $2176$
Sign $0.0490 + 0.998i$
Analytic cond. $233.843$
Root an. cond. $233.843$
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  + (0.831 − 0.555i)3-s + (−0.980 + 0.195i)5-s + (−0.382 − 0.923i)7-s + (0.382 − 0.923i)9-s + (−0.555 + 0.831i)11-s + (−0.980 − 0.195i)13-s + (−0.707 + 0.707i)15-s + (0.195 − 0.980i)19-s + (−0.831 − 0.555i)21-s + (−0.923 − 0.382i)23-s + (0.923 − 0.382i)25-s + (−0.195 − 0.980i)27-s + (−0.555 − 0.831i)29-s − i·31-s + i·33-s + ⋯
L(s)  = 1  + (0.831 − 0.555i)3-s + (−0.980 + 0.195i)5-s + (−0.382 − 0.923i)7-s + (0.382 − 0.923i)9-s + (−0.555 + 0.831i)11-s + (−0.980 − 0.195i)13-s + (−0.707 + 0.707i)15-s + (0.195 − 0.980i)19-s + (−0.831 − 0.555i)21-s + (−0.923 − 0.382i)23-s + (0.923 − 0.382i)25-s + (−0.195 − 0.980i)27-s + (−0.555 − 0.831i)29-s − i·31-s + i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2176 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0490 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2176 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0490 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2176\)    =    \(2^{7} \cdot 17\)
Sign: $0.0490 + 0.998i$
Analytic conductor: \(233.843\)
Root analytic conductor: \(233.843\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2176} (1971, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2176,\ (1:\ ),\ 0.0490 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2584970968 - 0.2461096954i\)
\(L(\frac12)\) \(\approx\) \(-0.2584970968 - 0.2461096954i\)
\(L(1)\) \(\approx\) \(0.7846528794 - 0.3841190706i\)
\(L(1)\) \(\approx\) \(0.7846528794 - 0.3841190706i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + (0.831 - 0.555i)T \)
5 \( 1 + (-0.980 + 0.195i)T \)
7 \( 1 + (-0.382 - 0.923i)T \)
11 \( 1 + (-0.555 + 0.831i)T \)
13 \( 1 + (-0.980 - 0.195i)T \)
19 \( 1 + (0.195 - 0.980i)T \)
23 \( 1 + (-0.923 - 0.382i)T \)
29 \( 1 + (-0.555 - 0.831i)T \)
31 \( 1 - iT \)
37 \( 1 + (-0.195 - 0.980i)T \)
41 \( 1 + (-0.923 - 0.382i)T \)
43 \( 1 + (-0.831 - 0.555i)T \)
47 \( 1 + (0.707 + 0.707i)T \)
53 \( 1 + (0.555 - 0.831i)T \)
59 \( 1 + (0.980 - 0.195i)T \)
61 \( 1 + (0.831 - 0.555i)T \)
67 \( 1 + (0.831 - 0.555i)T \)
71 \( 1 + (0.382 + 0.923i)T \)
73 \( 1 + (0.382 - 0.923i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + (-0.195 + 0.980i)T \)
89 \( 1 + (-0.923 + 0.382i)T \)
97 \( 1 - iT \)
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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

−20.00106113965406317468927430018, −19.5365090863917434417318432272, −18.69501523569747553695988127337, −18.43965752922221922015327696988, −16.93800091834288039237686088914, −16.21689626260608465107710011056, −15.86909207641843142170774983133, −15.043580215050529386243043666831, −14.5626393784706300661382914372, −13.6587710638822316247102065958, −12.81567250834308039701177058460, −12.06370948374300096526864201592, −11.463379319400987253944284454046, −10.35328967336020769493441438102, −9.83137755500191042640930413475, −8.798972143397173108731492706636, −8.40287525534018267593962508493, −7.68635021788897892070176093867, −6.82011147686465425318641951487, −5.52205386608620107050312405661, −5.01618793503277768459185315604, −3.927588243375459628684904193961, −3.28226137364164872308435556607, −2.61237313734121795405164888683, −1.53048799946450889266646755172, 0.07908297242550608745571223720, 0.61729838467944558809082530107, 2.0980132859766147710033518264, 2.69148731308102903829270527320, 3.782365182280982007822116223998, 4.20671126015315689179235949528, 5.27967842624089220119798481102, 6.744950212015348385238184269, 7.10260790545959299673783149914, 7.78141690290053862456398120273, 8.32298754121674157599868733526, 9.54585299281988830620766086358, 9.974524311905346905620614875200, 10.95446160423030301314831099541, 11.88144827662374899943338925617, 12.55564316415433071073357858912, 13.173294378838876187762371659312, 13.939053538519665633825362654389, 14.73275057060503830379242624840, 15.332695727718942749794914513485, 15.91850511279572776766324081625, 16.970543152204081507936413332086, 17.66219007089110304405464105949, 18.47886475766308228960783863121, 19.20188872220886509017187953146

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