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Properties

Label 1-2349-2349.61-r1-0-0
Degree $1$
Conductor $2349$
Sign $-0.378 + 0.925i$
Analytic cond. $252.435$
Root an. cond. $252.435$
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  + (0.629 + 0.776i)2-s + (−0.206 + 0.978i)4-s + (0.886 + 0.463i)5-s + (−0.950 − 0.310i)7-s + (−0.889 + 0.456i)8-s + (0.198 + 0.980i)10-s + (0.994 − 0.107i)11-s + (0.870 − 0.492i)13-s + (−0.357 − 0.933i)14-s + (−0.914 − 0.403i)16-s + (−0.984 − 0.173i)17-s + (0.478 − 0.878i)19-s + (−0.636 + 0.771i)20-s + (0.710 + 0.704i)22-s + (0.514 + 0.857i)23-s + ⋯
L(s)  = 1  + (0.629 + 0.776i)2-s + (−0.206 + 0.978i)4-s + (0.886 + 0.463i)5-s + (−0.950 − 0.310i)7-s + (−0.889 + 0.456i)8-s + (0.198 + 0.980i)10-s + (0.994 − 0.107i)11-s + (0.870 − 0.492i)13-s + (−0.357 − 0.933i)14-s + (−0.914 − 0.403i)16-s + (−0.984 − 0.173i)17-s + (0.478 − 0.878i)19-s + (−0.636 + 0.771i)20-s + (0.710 + 0.704i)22-s + (0.514 + 0.857i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2349\)    =    \(3^{4} \cdot 29\)
Sign: $-0.378 + 0.925i$
Analytic conductor: \(252.435\)
Root analytic conductor: \(252.435\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2349} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2349,\ (1:\ ),\ -0.378 + 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.969884547 + 2.933204507i\)
\(L(\frac12)\) \(\approx\) \(1.969884547 + 2.933204507i\)
\(L(1)\) \(\approx\) \(1.372253971 + 0.8586548528i\)
\(L(1)\) \(\approx\) \(1.372253971 + 0.8586548528i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.629 + 0.776i)T \)
5 \( 1 + (0.886 + 0.463i)T \)
7 \( 1 + (-0.950 - 0.310i)T \)
11 \( 1 + (0.994 - 0.107i)T \)
13 \( 1 + (0.870 - 0.492i)T \)
17 \( 1 + (-0.984 - 0.173i)T \)
19 \( 1 + (0.478 - 0.878i)T \)
23 \( 1 + (0.514 + 0.857i)T \)
31 \( 1 + (-0.924 + 0.380i)T \)
37 \( 1 + (0.992 - 0.124i)T \)
41 \( 1 + (-0.230 + 0.973i)T \)
43 \( 1 + (-0.999 - 0.0415i)T \)
47 \( 1 + (-0.278 - 0.960i)T \)
53 \( 1 + (0.0747 - 0.997i)T \)
59 \( 1 + (-0.993 + 0.116i)T \)
61 \( 1 + (0.978 - 0.206i)T \)
67 \( 1 + (0.556 + 0.830i)T \)
71 \( 1 + (-0.456 + 0.889i)T \)
73 \( 1 + (-0.246 - 0.969i)T \)
79 \( 1 + (0.999 - 0.00831i)T \)
83 \( 1 + (0.426 - 0.904i)T \)
89 \( 1 + (0.911 + 0.411i)T \)
97 \( 1 + (0.966 - 0.254i)T \)
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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

−19.28472121909511040508024778390, −18.56559247454619741154866604058, −18.0408527338215034210139936255, −16.919032878178182282071410055996, −16.38411565413005016173867324800, −15.50590445856325219637246193465, −14.643407340744990938224688946359, −13.94504191471797138436775725017, −13.32836472266946207264724204951, −12.70409381750737009063630477760, −12.117855748943382666352562260555, −11.22300452364974585617176016133, −10.49244736541464928630781263631, −9.571982273608221843689828358606, −9.21018799067966909509619244349, −8.53884236037796408207889028757, −6.86784267853816709221792857147, −6.237213309137920226466281463118, −5.80983826814033603162441172564, −4.7299860009803439218203931995, −3.98023876203138824094091106995, −3.20069738328078832690576479774, −2.18096201646836123920223995171, −1.5274887523326225929522548464, −0.57202493011046375141795227320, 0.80361106167067772905856309691, 2.10291818194941358721663320594, 3.24116728506130988513345435710, 3.52978064296000309668519700724, 4.70974609615796580268915706106, 5.56550153863045579940044813771, 6.37044598764507942162286672547, 6.727890986131677138630929694086, 7.454871612905508598085556433609, 8.72122206011892425960042782226, 9.21253050305147407869361647171, 9.95905358345042908070995713829, 11.09600977130089017604639029402, 11.60414736957983298031632700743, 12.88774899548826637321896609014, 13.309205651026926433132153353404, 13.701596320224622453870931298780, 14.65744107060711363077572379971, 15.23670140671784805096830871174, 16.08896072145964560909180717721, 16.63236409210163752613118314309, 17.492374742562991521577688925778, 17.91598589853795766056794958693, 18.72351456543656277353306626722, 19.85054869465022383280728774889

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