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Properties

Label 1-2349-2349.239-r1-0-0
Degree $1$
Conductor $2349$
Sign $-0.869 + 0.493i$
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  + (0.976 − 0.214i)2-s + (0.908 − 0.418i)4-s + (0.189 − 0.981i)5-s + (−0.0912 − 0.995i)7-s + (0.797 − 0.603i)8-s + (−0.0249 − 0.999i)10-s + (−0.334 − 0.942i)11-s + (0.997 + 0.0664i)13-s + (−0.302 − 0.953i)14-s + (0.649 − 0.760i)16-s + (−0.173 − 0.984i)17-s + (−0.661 − 0.749i)19-s + (−0.238 − 0.971i)20-s + (−0.528 − 0.848i)22-s + (−0.991 + 0.132i)23-s + ⋯
L(s)  = 1  + (0.976 − 0.214i)2-s + (0.908 − 0.418i)4-s + (0.189 − 0.981i)5-s + (−0.0912 − 0.995i)7-s + (0.797 − 0.603i)8-s + (−0.0249 − 0.999i)10-s + (−0.334 − 0.942i)11-s + (0.997 + 0.0664i)13-s + (−0.302 − 0.953i)14-s + (0.649 − 0.760i)16-s + (−0.173 − 0.984i)17-s + (−0.661 − 0.749i)19-s + (−0.238 − 0.971i)20-s + (−0.528 − 0.848i)22-s + (−0.991 + 0.132i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2349\)    =    \(3^{4} \cdot 29\)
Sign: $-0.869 + 0.493i$
Analytic conductor: \(252.435\)
Root analytic conductor: \(252.435\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2349} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2349,\ (1:\ ),\ -0.869 + 0.493i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.9269709155 - 3.510005036i\)
\(L(\frac12)\) \(\approx\) \(-0.9269709155 - 3.510005036i\)
\(L(1)\) \(\approx\) \(1.442054462 - 1.227415594i\)
\(L(1)\) \(\approx\) \(1.442054462 - 1.227415594i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.976 - 0.214i)T \)
5 \( 1 + (0.189 - 0.981i)T \)
7 \( 1 + (-0.0912 - 0.995i)T \)
11 \( 1 + (-0.334 - 0.942i)T \)
13 \( 1 + (0.997 + 0.0664i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (-0.661 - 0.749i)T \)
23 \( 1 + (-0.991 + 0.132i)T \)
31 \( 1 + (0.485 + 0.874i)T \)
37 \( 1 + (0.542 - 0.840i)T \)
41 \( 1 + (-0.973 + 0.230i)T \)
43 \( 1 + (-0.755 + 0.655i)T \)
47 \( 1 + (-0.986 + 0.165i)T \)
53 \( 1 + (-0.826 - 0.563i)T \)
59 \( 1 + (0.993 + 0.116i)T \)
61 \( 1 + (0.908 + 0.418i)T \)
67 \( 1 + (0.861 - 0.507i)T \)
71 \( 1 + (0.797 + 0.603i)T \)
73 \( 1 + (-0.411 - 0.911i)T \)
79 \( 1 + (-0.441 + 0.897i)T \)
83 \( 1 + (-0.786 + 0.616i)T \)
89 \( 1 + (0.969 + 0.246i)T \)
97 \( 1 + (0.999 + 0.0332i)T \)
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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

−20.022557951148115116981401140758, −18.96282130534568061063316631891, −18.51398063891121692247466137266, −17.655922932372768476986649922544, −16.93368877355240749870354850015, −15.837913880909948972665036484622, −15.39733052004421355491929003594, −14.81768935169057365682642590114, −14.2399296970431206514700103961, −13.23013322957461901935068630623, −12.79561737130433534108710230239, −11.85626322422480068709020070770, −11.35966001212453714384092033470, −10.34387000791143758332635297164, −9.928575647104004481468089477253, −8.44671821458670787668995531539, −8.02366937181849703610408906543, −6.92318140774125333662938478498, −6.1823363576550000917998211358, −5.891430075174694178051083798741, −4.8010450447022680736505144690, −3.86534622255545702087653695543, −3.223197498250465826407570675877, −2.1151454121739333388806816601, −1.86957400776793145978579712465, 0.386668803626972099554900278284, 1.06851075591772862598437677601, 2.03297888817237327677223048574, 3.16007478407284638909692708891, 3.86910887181102298572278198820, 4.66166234043054094021286232889, 5.28543444577204060463664125626, 6.229040575080048503050301651987, 6.79820716974435973371697833847, 7.91709643886821746300664962582, 8.55610590549226838006256270560, 9.63573660285042400625257974503, 10.35519424355440900617864590502, 11.25310100891016172822445254044, 11.617788322306972981282425848813, 12.82526487456453586053145599048, 13.20514224451750013489161343996, 13.81260298324276301645382379219, 14.31924184848939889506940319637, 15.579018114106499459124592682503, 16.167813983067616269528862729138, 16.4587473864298026298352820725, 17.45425591542812228092370597210, 18.29643231851536081533226939400, 19.36614994659911999604338651049

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