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Properties

Label 1-2349-2349.1525-r1-0-0
Degree $1$
Conductor $2349$
Sign $-0.290 + 0.956i$
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  + (−0.918 − 0.396i)2-s + (0.686 + 0.727i)4-s + (0.0581 + 0.998i)5-s + (0.973 + 0.230i)7-s + (−0.342 − 0.939i)8-s + (0.342 − 0.939i)10-s + (−0.549 + 0.835i)11-s + (0.993 + 0.116i)13-s + (−0.802 − 0.597i)14-s + (−0.0581 + 0.998i)16-s + (−0.642 − 0.766i)17-s + (0.642 − 0.766i)19-s + (−0.686 + 0.727i)20-s + (0.835 − 0.549i)22-s + (0.973 − 0.230i)23-s + ⋯
L(s)  = 1  + (−0.918 − 0.396i)2-s + (0.686 + 0.727i)4-s + (0.0581 + 0.998i)5-s + (0.973 + 0.230i)7-s + (−0.342 − 0.939i)8-s + (0.342 − 0.939i)10-s + (−0.549 + 0.835i)11-s + (0.993 + 0.116i)13-s + (−0.802 − 0.597i)14-s + (−0.0581 + 0.998i)16-s + (−0.642 − 0.766i)17-s + (0.642 − 0.766i)19-s + (−0.686 + 0.727i)20-s + (0.835 − 0.549i)22-s + (0.973 − 0.230i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2349\)    =    \(3^{4} \cdot 29\)
Sign: $-0.290 + 0.956i$
Analytic conductor: \(252.435\)
Root analytic conductor: \(252.435\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2349} (1525, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2349,\ (1:\ ),\ -0.290 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8036430087 + 1.083628130i\)
\(L(\frac12)\) \(\approx\) \(0.8036430087 + 1.083628130i\)
\(L(1)\) \(\approx\) \(0.8059569784 + 0.1362081721i\)
\(L(1)\) \(\approx\) \(0.8059569784 + 0.1362081721i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.918 - 0.396i)T \)
5 \( 1 + (0.0581 + 0.998i)T \)
7 \( 1 + (0.973 + 0.230i)T \)
11 \( 1 + (-0.549 + 0.835i)T \)
13 \( 1 + (0.993 + 0.116i)T \)
17 \( 1 + (-0.642 - 0.766i)T \)
19 \( 1 + (0.642 - 0.766i)T \)
23 \( 1 + (0.973 - 0.230i)T \)
31 \( 1 + (0.957 + 0.286i)T \)
37 \( 1 + (-0.984 + 0.173i)T \)
41 \( 1 + (-0.918 + 0.396i)T \)
43 \( 1 + (-0.448 - 0.893i)T \)
47 \( 1 + (0.957 - 0.286i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.835 + 0.549i)T \)
61 \( 1 + (0.727 + 0.686i)T \)
67 \( 1 + (-0.597 + 0.802i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (-0.342 - 0.939i)T \)
79 \( 1 + (-0.918 - 0.396i)T \)
83 \( 1 + (0.396 - 0.918i)T \)
89 \( 1 + (0.342 + 0.939i)T \)
97 \( 1 + (-0.998 - 0.0581i)T \)
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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

−19.01012099722332983525533309396, −18.47033446878100426404978226110, −17.621969247075815278373987827844, −17.10996543607900951590584249054, −16.49855496798253158121659888600, −15.631677473132844226540906871260, −15.313600633522667640877580294603, −14.05169627024087297206009829303, −13.66282977799989088576316737128, −12.65119859654073375064464224406, −11.68864886432223722250201892098, −11.02562827596070820054685120097, −10.48030006983627366969072566923, −9.48513399144159618223251963196, −8.61671096393427921050903926427, −8.29980408715793384770853317203, −7.68052515522169065278872683593, −6.56326638075016308726875013428, −5.69632136716476293281129103040, −5.186924677928715517750361549024, −4.19890236069619083916779351252, −3.05440370250983796159589599652, −1.72147545890081753622630283247, −1.262058084076318639646538905683, −0.35191346401602109340026231803, 0.93623346197114386949594930621, 1.91495762486903819271967177400, 2.608294201887188151415369677, 3.35050127941698647665355272198, 4.48928683663279763386392064304, 5.377395613833200370263955596956, 6.65968325492054385199618204614, 7.05513071164992200971734615777, 7.8502133210429961782962290414, 8.67362643545572331850083690399, 9.306672000113289474072821586831, 10.38673969604537017247146316641, 10.682647321715464133720371262302, 11.639176938882568047936867065933, 11.84753191340883352242924883384, 13.19403603151233972344025149494, 13.73845888138650933888137384447, 14.81096137947374393589902983337, 15.50978487909302586743519198780, 15.86288813303189517529278032148, 17.16548343390028417820234600006, 17.67193767166199830431770841944, 18.279739183462058203533570161, 18.64112874649306967083624871260, 19.49278018782484551725680966454

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