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Properties

Label 1-2349-2349.470-r1-0-0
Degree $1$
Conductor $2349$
Sign $0.912 + 0.409i$
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  + (0.238 − 0.971i)2-s + (−0.886 − 0.463i)4-s + (0.870 − 0.492i)5-s + (0.0415 − 0.999i)7-s + (−0.661 + 0.749i)8-s + (−0.270 − 0.962i)10-s + (0.426 − 0.904i)11-s + (−0.950 + 0.310i)13-s + (−0.960 − 0.278i)14-s + (0.570 + 0.821i)16-s + (−0.939 + 0.342i)17-s + (−0.995 + 0.0995i)19-s + (−0.999 + 0.0332i)20-s + (−0.776 − 0.629i)22-s + (−0.807 − 0.590i)23-s + ⋯
L(s)  = 1  + (0.238 − 0.971i)2-s + (−0.886 − 0.463i)4-s + (0.870 − 0.492i)5-s + (0.0415 − 0.999i)7-s + (−0.661 + 0.749i)8-s + (−0.270 − 0.962i)10-s + (0.426 − 0.904i)11-s + (−0.950 + 0.310i)13-s + (−0.960 − 0.278i)14-s + (0.570 + 0.821i)16-s + (−0.939 + 0.342i)17-s + (−0.995 + 0.0995i)19-s + (−0.999 + 0.0332i)20-s + (−0.776 − 0.629i)22-s + (−0.807 − 0.590i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2349\)    =    \(3^{4} \cdot 29\)
Sign: $0.912 + 0.409i$
Analytic conductor: \(252.435\)
Root analytic conductor: \(252.435\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2349} (470, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2349,\ (1:\ ),\ 0.912 + 0.409i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1900815618 - 0.04065925060i\)
\(L(\frac12)\) \(\approx\) \(-0.1900815618 - 0.04065925060i\)
\(L(1)\) \(\approx\) \(0.6427894490 - 0.7330420614i\)
\(L(1)\) \(\approx\) \(0.6427894490 - 0.7330420614i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.238 - 0.971i)T \)
5 \( 1 + (0.870 - 0.492i)T \)
7 \( 1 + (0.0415 - 0.999i)T \)
11 \( 1 + (0.426 - 0.904i)T \)
13 \( 1 + (-0.950 + 0.310i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (-0.995 + 0.0995i)T \)
23 \( 1 + (-0.807 - 0.590i)T \)
31 \( 1 + (-0.334 - 0.942i)T \)
37 \( 1 + (0.0249 - 0.999i)T \)
41 \( 1 + (0.893 - 0.448i)T \)
43 \( 1 + (-0.861 + 0.507i)T \)
47 \( 1 + (-0.710 + 0.704i)T \)
53 \( 1 + (-0.955 - 0.294i)T \)
59 \( 1 + (-0.973 - 0.230i)T \)
61 \( 1 + (0.886 - 0.463i)T \)
67 \( 1 + (-0.816 + 0.576i)T \)
71 \( 1 + (0.661 + 0.749i)T \)
73 \( 1 + (0.998 + 0.0498i)T \)
79 \( 1 + (-0.206 - 0.978i)T \)
83 \( 1 + (0.999 + 0.0166i)T \)
89 \( 1 + (0.921 - 0.388i)T \)
97 \( 1 + (0.157 + 0.987i)T \)
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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

−19.98768906110928413863170358867, −19.23136461907012506946034851877, −18.19221005228544855358623588048, −17.93527784661556267709097648400, −17.27192299711101550283173603047, −16.58571165788577191171990604026, −15.441301460334672501409468974008, −15.1516724952456768781100281854, −14.481035187006635981103485918708, −13.78541288605475450398611071873, −12.91158292561089402926438733427, −12.38133273788197281893271487732, −11.543748468279225170724392522837, −10.340755432797432015514253835295, −9.60569600463051988441241282821, −9.109057817983560891405234744815, −8.251092895831949735884367376819, −7.30953995122700287503870230981, −6.57772356937928556881235368393, −6.09173650586150743434214665557, −5.07587297037299750958045088968, −4.6782859080416164964145338030, −3.41639410947215367971494981552, −2.463305362403226927536177817478, −1.756522818778491071281319599628, 0.03516705773258495263755690612, 0.746634225997485780879904258357, 1.84478765104724215498314374967, 2.34211113335621485566875768092, 3.573474399050343909162899687323, 4.33615885140122863962358707058, 4.86839034575548535382455507994, 6.028876456602472775180755016294, 6.47946073551832678754480013410, 7.8325004068830541203177624786, 8.668888315547980213462708736975, 9.37303586115514315597039208911, 10.01686691323023784437077095035, 10.78483108618793188880581876751, 11.30126320536048231558531602921, 12.36043565328079831640993887266, 12.97425554643305013447822854506, 13.52348794221037741952492255731, 14.35947841697116110853342395353, 14.60423362615910191693881909515, 16.05988978762876989762110344652, 16.87888769994444215568312374060, 17.3452962732183693233221511963, 17.98754887565463578863977916630, 19.00712149884274769627826916351

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