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Properties

Label 1-2349-2349.1060-r0-0-0
Degree $1$
Conductor $2349$
Sign $0.787 - 0.615i$
Analytic cond. $10.9087$
Root an. cond. $10.9087$
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  + (−0.927 + 0.373i)2-s + (0.721 − 0.692i)4-s + (0.964 − 0.262i)5-s + (−0.960 + 0.278i)7-s + (−0.411 + 0.911i)8-s + (−0.797 + 0.603i)10-s + (−0.886 + 0.463i)11-s + (0.933 − 0.357i)13-s + (0.786 − 0.616i)14-s + (0.0415 − 0.999i)16-s + (0.173 − 0.984i)17-s + (−0.998 − 0.0498i)19-s + (0.514 − 0.857i)20-s + (0.649 − 0.760i)22-s + (0.744 + 0.667i)23-s + ⋯
L(s)  = 1  + (−0.927 + 0.373i)2-s + (0.721 − 0.692i)4-s + (0.964 − 0.262i)5-s + (−0.960 + 0.278i)7-s + (−0.411 + 0.911i)8-s + (−0.797 + 0.603i)10-s + (−0.886 + 0.463i)11-s + (0.933 − 0.357i)13-s + (0.786 − 0.616i)14-s + (0.0415 − 0.999i)16-s + (0.173 − 0.984i)17-s + (−0.998 − 0.0498i)19-s + (0.514 − 0.857i)20-s + (0.649 − 0.760i)22-s + (0.744 + 0.667i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2349\)    =    \(3^{4} \cdot 29\)
Sign: $0.787 - 0.615i$
Analytic conductor: \(10.9087\)
Root analytic conductor: \(10.9087\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2349} (1060, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2349,\ (0:\ ),\ 0.787 - 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9074831747 - 0.3126282607i\)
\(L(\frac12)\) \(\approx\) \(0.9074831747 - 0.3126282607i\)
\(L(1)\) \(\approx\) \(0.7458300879 + 0.01609137210i\)
\(L(1)\) \(\approx\) \(0.7458300879 + 0.01609137210i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.927 + 0.373i)T \)
5 \( 1 + (0.964 - 0.262i)T \)
7 \( 1 + (-0.960 + 0.278i)T \)
11 \( 1 + (-0.886 + 0.463i)T \)
13 \( 1 + (0.933 - 0.357i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (-0.998 - 0.0498i)T \)
23 \( 1 + (0.744 + 0.667i)T \)
31 \( 1 + (0.908 + 0.418i)T \)
37 \( 1 + (0.698 - 0.715i)T \)
41 \( 1 + (-0.286 - 0.957i)T \)
43 \( 1 + (-0.710 + 0.704i)T \)
47 \( 1 + (-0.610 - 0.792i)T \)
53 \( 1 + (-0.988 + 0.149i)T \)
59 \( 1 + (0.597 + 0.802i)T \)
61 \( 1 + (0.721 + 0.692i)T \)
67 \( 1 + (-0.976 + 0.214i)T \)
71 \( 1 + (-0.411 - 0.911i)T \)
73 \( 1 + (-0.0249 - 0.999i)T \)
79 \( 1 + (-0.157 - 0.987i)T \)
83 \( 1 + (-0.870 - 0.492i)T \)
89 \( 1 + (0.980 + 0.198i)T \)
97 \( 1 + (-0.983 + 0.181i)T \)
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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

−19.41140079608596272877066346922, −18.7952351613701168373229016895, −18.515619425051817789946021728279, −17.4687490329756263781075923405, −16.93303838038830150183350496064, −16.34602052461399007173301507849, −15.57369934020903732595300429160, −14.7567088989936150567017706969, −13.620385073331204634032619058973, −13.00699446691875027601472646633, −12.65583700082176338508561400096, −11.28428194262741919464131414120, −10.80011820869664404755451447033, −10.062094370394302975550696562631, −9.65167112409240931144761422801, −8.53939003402392831688651531389, −8.24552536981306293833929062052, −6.913297532908722597813905844132, −6.387221539466539793203178610094, −5.85493879829812044938712718746, −4.4349041522411748952147771567, −3.348546770985147618740762000630, −2.78129885053395857878769113461, −1.86433946373217665074010369694, −0.9091433783047085378646959972, 0.51045720185844580647516169997, 1.611881515964093972121287313360, 2.5236793378000061953214026711, 3.17468351480758065732668642370, 4.76627898775109106947368591156, 5.53606049833153399934723607227, 6.17919067792795730615971079000, 6.85813809000542438706259600266, 7.70642178509226571765150495276, 8.72926906082383438965248900419, 9.12193750835096648213884533003, 10.05409290038826611593379339419, 10.344200196809573973048254246845, 11.299895842657956470722434514493, 12.301627466009808090944103267152, 13.184240485218124221813756744300, 13.57414671625338694259425986287, 14.73214548671816519927200609691, 15.40925162723110378984796102591, 16.13308174804953345929451384391, 16.568317043010992830035220558621, 17.568773904480351934203218908980, 17.96817961671683785022752818178, 18.68445308152038052473065550718, 19.32323119465303144681349399104

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