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Properties

Label 1-2349-2349.110-r1-0-0
Degree $1$
Conductor $2349$
Sign $0.319 - 0.947i$
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  + (−0.807 + 0.590i)2-s + (0.302 − 0.953i)4-s + (−0.986 + 0.165i)5-s + (−0.976 − 0.214i)7-s + (0.318 + 0.947i)8-s + (0.698 − 0.715i)10-s + (0.0912 − 0.995i)11-s + (−0.528 + 0.848i)13-s + (0.914 − 0.403i)14-s + (−0.816 − 0.576i)16-s + (−0.766 − 0.642i)17-s + (−0.411 + 0.911i)19-s + (−0.140 + 0.990i)20-s + (0.514 + 0.857i)22-s + (0.441 − 0.897i)23-s + ⋯
L(s)  = 1  + (−0.807 + 0.590i)2-s + (0.302 − 0.953i)4-s + (−0.986 + 0.165i)5-s + (−0.976 − 0.214i)7-s + (0.318 + 0.947i)8-s + (0.698 − 0.715i)10-s + (0.0912 − 0.995i)11-s + (−0.528 + 0.848i)13-s + (0.914 − 0.403i)14-s + (−0.816 − 0.576i)16-s + (−0.766 − 0.642i)17-s + (−0.411 + 0.911i)19-s + (−0.140 + 0.990i)20-s + (0.514 + 0.857i)22-s + (0.441 − 0.897i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2349\)    =    \(3^{4} \cdot 29\)
Sign: $0.319 - 0.947i$
Analytic conductor: \(252.435\)
Root analytic conductor: \(252.435\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2349} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2349,\ (1:\ ),\ 0.319 - 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1456022167 - 0.1045375447i\)
\(L(\frac12)\) \(\approx\) \(0.1456022167 - 0.1045375447i\)
\(L(1)\) \(\approx\) \(0.4407095776 + 0.1011592597i\)
\(L(1)\) \(\approx\) \(0.4407095776 + 0.1011592597i\)

Euler product

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

−19.46278612635595589281350982275, −19.126946309408859288768621507616, −18.19882407070167083372314391175, −17.37321803772067844996267597090, −16.95406235537412929825610873296, −15.90837326330183875111115742229, −15.42098408313770152253537529235, −14.93662917272224702889836442867, −13.324178528862925182066637021982, −12.763462167631918949557288177756, −12.383926385646992187356206340042, −11.43214635750765122482981148377, −10.861915005406323882703393787841, −9.945029326443602670633793953218, −9.35549073888035742658538948559, −8.62555382323071887780363563517, −7.786377545942491202872057466911, −7.12735594227707293393645842642, −6.4987855109421901447424651536, −5.12285369036149231711998410619, −4.202678497180228190694800410720, −3.45803105404719520774823453965, −2.69142293738905215063350844583, −1.77739386713748845653342507163, −0.47047996130922276364144166954, 0.09311092312424554961552491145, 0.9791675695869056195389697900, 2.3105868382323452553546508197, 3.22883812065488280636745849361, 4.17522540008809407032483321266, 5.043357210242627922888950566591, 6.2280210297744351928501800274, 6.69053999401412131274214200955, 7.36275515034890749607883684907, 8.255871911530831032495503159928, 8.87095562245420646684434278784, 9.598980496639566600972680078364, 10.44681495660966118629061774299, 11.1718384360600917928042810188, 11.76257195175753043463631989937, 12.75524041039711716735773599478, 13.638689206731374458013760302976, 14.56176696033222504878945653138, 14.97251157122324908495385757652, 16.074344765425405499426512024701, 16.43662734746838438429526531843, 16.7032242511264006523174988005, 17.99778704164050669600985806578, 18.65940735754968640742334402146, 19.17778720290984249451360107167

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