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Properties

Label 1-2944-2944.459-r0-0-0
Degree $1$
Conductor $2944$
Sign $-0.0490 + 0.998i$
Analytic cond. $13.6718$
Root an. cond. $13.6718$
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  + (0.980 − 0.195i)3-s + (−0.555 − 0.831i)5-s + (−0.923 − 0.382i)7-s + (0.923 − 0.382i)9-s + (−0.195 + 0.980i)11-s + (−0.555 + 0.831i)13-s + (−0.707 − 0.707i)15-s + (0.707 − 0.707i)17-s + (−0.831 − 0.555i)19-s + (−0.980 − 0.195i)21-s + (−0.382 + 0.923i)25-s + (0.831 − 0.555i)27-s + (0.195 + 0.980i)29-s − i·31-s − i·33-s + ⋯
L(s)  = 1  + (0.980 − 0.195i)3-s + (−0.555 − 0.831i)5-s + (−0.923 − 0.382i)7-s + (0.923 − 0.382i)9-s + (−0.195 + 0.980i)11-s + (−0.555 + 0.831i)13-s + (−0.707 − 0.707i)15-s + (0.707 − 0.707i)17-s + (−0.831 − 0.555i)19-s + (−0.980 − 0.195i)21-s + (−0.382 + 0.923i)25-s + (0.831 − 0.555i)27-s + (0.195 + 0.980i)29-s − i·31-s − i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2944 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0490 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2944 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0490 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2944\)    =    \(2^{7} \cdot 23\)
Sign: $-0.0490 + 0.998i$
Analytic conductor: \(13.6718\)
Root analytic conductor: \(13.6718\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2944} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2944,\ (0:\ ),\ -0.0490 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4924431256 + 0.5172291897i\)
\(L(\frac12)\) \(\approx\) \(0.4924431256 + 0.5172291897i\)
\(L(1)\) \(\approx\) \(0.9897054266 - 0.1122856644i\)
\(L(1)\) \(\approx\) \(0.9897054266 - 0.1122856644i\)

Euler product

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

−19.183179243805643220628840197, −18.56209367797308398896114246456, −17.659394806106823309481522418050, −16.52745371707701172098110909311, −16.05427117526407235630076676009, −15.220498100627764978543916572001, −14.86214383695077974113992655175, −14.13654163851046055685240599573, −13.30554643457218154928926837368, −12.67046025431514012765263628266, −11.957890921060235660865395932434, −10.91224011764000228582623031394, −10.11587002957466891048264789252, −9.9192242191685539044306372059, −8.57996079954563790036979079512, −8.28469769936487928507681787747, −7.48873839788508449039716390382, −6.60844467940795357973099472611, −5.95240237784981783110582468982, −4.90129215994990364166706957056, −3.689168759164330471295928610915, −3.35528106355937753290925528102, −2.741830188291870224094892494662, −1.78846409673070541831059883138, −0.19053845570086572996362369513, 1.12015920230078012811753715672, 2.05595458806705817362241665990, 2.927402471506275202612447330052, 3.781581400344278216767716596347, 4.46023023460505929281365530363, 5.11835464351248217623591736745, 6.51645721165023186385640180926, 7.164343660091313381322861579702, 7.64370426626326969058981005727, 8.61540925816811156576106979879, 9.25389243904652230210741112167, 9.75635979093441629754873398433, 10.5284024099191767906631321529, 11.82355747214595393655419949495, 12.34368429972204540049836382969, 12.944279174954982329913805103990, 13.57981082471769608582667638784, 14.33239919314809159266302569095, 15.20495146202324719836003781948, 15.63115668522273399951968294782, 16.54004879120495977905301278937, 16.949682453637710562006210421061, 17.99335258320312454327845190864, 18.945301752601104821864437996401, 19.29137254342675871446595720091

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