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Properties

Label 1-2349-2349.301-r1-0-0
Degree $1$
Conductor $2349$
Sign $-0.961 - 0.275i$
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  + (−0.821 − 0.570i)2-s + (0.349 + 0.936i)4-s + (0.470 − 0.882i)5-s + (0.986 − 0.165i)7-s + (0.246 − 0.969i)8-s + (−0.889 + 0.456i)10-s + (−0.981 + 0.189i)11-s + (−0.302 − 0.953i)13-s + (−0.904 − 0.426i)14-s + (−0.755 + 0.655i)16-s + (0.984 − 0.173i)17-s + (0.388 − 0.921i)19-s + (0.991 + 0.132i)20-s + (0.914 + 0.403i)22-s + (−0.816 − 0.576i)23-s + ⋯
L(s)  = 1  + (−0.821 − 0.570i)2-s + (0.349 + 0.936i)4-s + (0.470 − 0.882i)5-s + (0.986 − 0.165i)7-s + (0.246 − 0.969i)8-s + (−0.889 + 0.456i)10-s + (−0.981 + 0.189i)11-s + (−0.302 − 0.953i)13-s + (−0.904 − 0.426i)14-s + (−0.755 + 0.655i)16-s + (0.984 − 0.173i)17-s + (0.388 − 0.921i)19-s + (0.991 + 0.132i)20-s + (0.914 + 0.403i)22-s + (−0.816 − 0.576i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2349\)    =    \(3^{4} \cdot 29\)
Sign: $-0.961 - 0.275i$
Analytic conductor: \(252.435\)
Root analytic conductor: \(252.435\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2349} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2349,\ (1:\ ),\ -0.961 - 0.275i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2384684226 - 1.699887518i\)
\(L(\frac12)\) \(\approx\) \(0.2384684226 - 1.699887518i\)
\(L(1)\) \(\approx\) \(0.7464412782 - 0.4969849654i\)
\(L(1)\) \(\approx\) \(0.7464412782 - 0.4969849654i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.821 - 0.570i)T \)
5 \( 1 + (0.470 - 0.882i)T \)
7 \( 1 + (0.986 - 0.165i)T \)
11 \( 1 + (-0.981 + 0.189i)T \)
13 \( 1 + (-0.302 - 0.953i)T \)
17 \( 1 + (0.984 - 0.173i)T \)
19 \( 1 + (0.388 - 0.921i)T \)
23 \( 1 + (-0.816 - 0.576i)T \)
31 \( 1 + (0.978 - 0.206i)T \)
37 \( 1 + (0.0995 + 0.995i)T \)
41 \( 1 + (-0.957 - 0.286i)T \)
43 \( 1 + (0.848 + 0.528i)T \)
47 \( 1 + (-0.0166 - 0.999i)T \)
53 \( 1 + (0.365 - 0.930i)T \)
59 \( 1 + (0.597 - 0.802i)T \)
61 \( 1 + (0.936 + 0.349i)T \)
67 \( 1 + (0.776 - 0.629i)T \)
71 \( 1 + (0.969 - 0.246i)T \)
73 \( 1 + (0.198 + 0.980i)T \)
79 \( 1 + (0.738 - 0.674i)T \)
83 \( 1 + (0.997 - 0.0664i)T \)
89 \( 1 + (0.999 + 0.0249i)T \)
97 \( 1 + (-0.590 + 0.807i)T \)
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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

−19.329045111141924403648125895744, −18.84719529463149155067381338353, −18.244322389123767975788321560, −17.717254528920551475817763365005, −17.00162938141014762662611098759, −16.21336115467811379408235071537, −15.47764738541401396927651401454, −14.72117034516681131579573299225, −14.1014386249399690260724555524, −13.76683512249871959296351294939, −12.261264586464621548388459404460, −11.53709458806846620458877531117, −10.80045941404754652187975030640, −10.13852116859805800512927465683, −9.59496132609828712760714489966, −8.57754767764771290816714630355, −7.75535489050348006684217258572, −7.43497574113502069815277628393, −6.35959271631064896369136688877, −5.65632952543459888405466675460, −5.083238126249139560869430053745, −3.839425230909039388087177559635, −2.567836082576646795860548405791, −1.952405643614707273194620473, −1.05985228482717835919679216119, 0.456238362901811167015207478592, 0.9696089657266098158862382377, 2.06691507649571629097777543093, 2.66728678410496226384571514317, 3.80254318394812865950547520201, 4.950178225537275810942483809016, 5.252437931772679803788921562062, 6.553808780894575418026129026414, 7.687795925107932308011584224510, 8.08410521905397711046423593926, 8.6592421390159061056757673774, 9.84061420390413894362816358397, 10.07400283094409343723767130286, 10.95700758747756420105395682781, 11.858984882028983974612791928724, 12.368193905292975291369860805, 13.225299027245129681921423172552, 13.749882956675075565101832431899, 14.899392395481874428636240733, 15.72700899037425971622055778435, 16.38967709066472743583348479684, 17.237132517515139000265390461706, 17.66934337063007176771321077577, 18.233928138255753614767509047460, 19.00778730660939039235344775469

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