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Properties

Label 1-2349-2349.1001-r0-0-0
Degree $1$
Conductor $2349$
Sign $-0.874 - 0.485i$
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  + (−0.590 − 0.807i)2-s + (−0.302 + 0.953i)4-s + (0.986 − 0.165i)5-s + (−0.976 − 0.214i)7-s + (0.947 − 0.318i)8-s + (−0.715 − 0.698i)10-s + (0.995 + 0.0912i)11-s + (0.528 − 0.848i)13-s + (0.403 + 0.914i)14-s + (−0.816 − 0.576i)16-s + (0.642 − 0.766i)17-s + (−0.911 − 0.411i)19-s + (−0.140 + 0.990i)20-s + (−0.514 − 0.857i)22-s + (0.441 − 0.897i)23-s + ⋯
L(s)  = 1  + (−0.590 − 0.807i)2-s + (−0.302 + 0.953i)4-s + (0.986 − 0.165i)5-s + (−0.976 − 0.214i)7-s + (0.947 − 0.318i)8-s + (−0.715 − 0.698i)10-s + (0.995 + 0.0912i)11-s + (0.528 − 0.848i)13-s + (0.403 + 0.914i)14-s + (−0.816 − 0.576i)16-s + (0.642 − 0.766i)17-s + (−0.911 − 0.411i)19-s + (−0.140 + 0.990i)20-s + (−0.514 − 0.857i)22-s + (0.441 − 0.897i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2349\)    =    \(3^{4} \cdot 29\)
Sign: $-0.874 - 0.485i$
Analytic conductor: \(10.9087\)
Root analytic conductor: \(10.9087\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2349} (1001, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2349,\ (0:\ ),\ -0.874 - 0.485i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2858579882 - 1.103176932i\)
\(L(\frac12)\) \(\approx\) \(0.2858579882 - 1.103176932i\)
\(L(1)\) \(\approx\) \(0.7321595984 - 0.4748285684i\)
\(L(1)\) \(\approx\) \(0.7321595984 - 0.4748285684i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.590 - 0.807i)T \)
5 \( 1 + (0.986 - 0.165i)T \)
7 \( 1 + (-0.976 - 0.214i)T \)
11 \( 1 + (0.995 + 0.0912i)T \)
13 \( 1 + (0.528 - 0.848i)T \)
17 \( 1 + (0.642 - 0.766i)T \)
19 \( 1 + (-0.911 - 0.411i)T \)
23 \( 1 + (0.441 - 0.897i)T \)
31 \( 1 + (-0.492 - 0.870i)T \)
37 \( 1 + (-0.478 - 0.878i)T \)
41 \( 1 + (0.918 + 0.396i)T \)
43 \( 1 + (-0.771 - 0.636i)T \)
47 \( 1 + (-0.821 + 0.570i)T \)
53 \( 1 + (-0.955 - 0.294i)T \)
59 \( 1 + (0.835 + 0.549i)T \)
61 \( 1 + (-0.953 + 0.302i)T \)
67 \( 1 + (0.254 - 0.966i)T \)
71 \( 1 + (-0.318 + 0.947i)T \)
73 \( 1 + (-0.840 - 0.542i)T \)
79 \( 1 + (-0.882 - 0.470i)T \)
83 \( 1 + (0.755 + 0.655i)T \)
89 \( 1 + (0.992 + 0.124i)T \)
97 \( 1 + (0.874 - 0.485i)T \)
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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

−19.4475657073471697091100136045, −19.12883171516683974124152303483, −18.481565827693198938761529134341, −17.54646857771914007521330945038, −17.00870666506750885972365249148, −16.477354877133025716244365314483, −15.7751866540882834104002069600, −14.720787766724547586919123156938, −14.41050887321811579524276886033, −13.494397738564607961202826276, −12.94082379813845461244826346874, −11.88962982175785031555424457244, −10.854300846261030071006814928973, −10.17629246817084859367653014762, −9.44307562492934703719065755239, −9.01424130066141789713605739285, −8.24311477240918876456679457141, −7.015705829966018851686531540931, −6.47238408951931041966815449499, −6.04161956093477439655518572607, −5.21581927091965312124153228535, −4.06819848260170818731757134844, −3.18565611347215782745324206899, −1.79082221407686333734529357701, −1.34401820071516896988680845402, 0.47879773199367774221260091260, 1.34042972289230821510483441041, 2.373656034502253343603329451993, 3.08932940572075955592051418565, 3.9046505032560501432660934254, 4.858429415149426878659716605043, 5.974695292710402387269867264077, 6.64205335980563394584053874101, 7.48816894908688082017712421262, 8.59299331934778122957766425332, 9.18532468599141651215074995788, 9.739586489872991535081282427826, 10.453446415282304044532896392159, 11.07178184764293142185160490070, 12.10779211613769651917958084782, 12.83166270878684469991772593867, 13.2032961830584099547681673467, 14.043060571122500923812933867799, 14.86646347628555462831051289250, 16.11797480759094382995941653320, 16.59152179030338073548582461224, 17.27581923642828058371228240374, 17.8616681780876867595158309630, 18.6787687598925421688490717668, 19.25446194945453956471528397040

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