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Properties

Label 1-2365-2365.1737-r0-0-0
Degree $1$
Conductor $2365$
Sign $-0.993 + 0.110i$
Analytic cond. $10.9830$
Root an. cond. $10.9830$
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  + (0.433 + 0.900i)2-s + (−0.563 − 0.826i)3-s + (−0.623 + 0.781i)4-s + (0.5 − 0.866i)6-s + (−0.866 + 0.5i)7-s + (−0.974 − 0.222i)8-s + (−0.365 + 0.930i)9-s + (0.997 + 0.0747i)12-s + (0.294 + 0.955i)13-s + (−0.826 − 0.563i)14-s + (−0.222 − 0.974i)16-s + (0.680 + 0.733i)17-s + (−0.997 + 0.0747i)18-s + (0.365 + 0.930i)19-s + (0.900 + 0.433i)21-s + ⋯
L(s)  = 1  + (0.433 + 0.900i)2-s + (−0.563 − 0.826i)3-s + (−0.623 + 0.781i)4-s + (0.5 − 0.866i)6-s + (−0.866 + 0.5i)7-s + (−0.974 − 0.222i)8-s + (−0.365 + 0.930i)9-s + (0.997 + 0.0747i)12-s + (0.294 + 0.955i)13-s + (−0.826 − 0.563i)14-s + (−0.222 − 0.974i)16-s + (0.680 + 0.733i)17-s + (−0.997 + 0.0747i)18-s + (0.365 + 0.930i)19-s + (0.900 + 0.433i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2365 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2365 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2365\)    =    \(5 \cdot 11 \cdot 43\)
Sign: $-0.993 + 0.110i$
Analytic conductor: \(10.9830\)
Root analytic conductor: \(10.9830\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2365} (1737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2365,\ (0:\ ),\ -0.993 + 0.110i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05526561878 + 0.9962683328i\)
\(L(\frac12)\) \(\approx\) \(0.05526561878 + 0.9962683328i\)
\(L(1)\) \(\approx\) \(0.7451970269 + 0.4897023508i\)
\(L(1)\) \(\approx\) \(0.7451970269 + 0.4897023508i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.433 + 0.900i)T \)
3 \( 1 + (-0.563 - 0.826i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (0.294 + 0.955i)T \)
17 \( 1 + (0.680 + 0.733i)T \)
19 \( 1 + (0.365 + 0.930i)T \)
23 \( 1 + (0.149 + 0.988i)T \)
29 \( 1 + (0.826 + 0.563i)T \)
31 \( 1 + (0.0747 - 0.997i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (0.781 + 0.623i)T \)
53 \( 1 + (0.294 - 0.955i)T \)
59 \( 1 + (0.222 + 0.974i)T \)
61 \( 1 + (-0.0747 - 0.997i)T \)
67 \( 1 + (-0.930 + 0.365i)T \)
71 \( 1 + (-0.988 - 0.149i)T \)
73 \( 1 + (0.294 + 0.955i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.563 + 0.826i)T \)
89 \( 1 + (-0.826 + 0.563i)T \)
97 \( 1 + (-0.781 + 0.623i)T \)
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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

−19.50395385750908534215973420535, −18.54839260162728281632973465465, −17.850550578606829821718514359, −17.171469739722392629394585810881, −16.206363767323749697141558574680, −15.682709133514192954340522423149, −14.941286546680977936246833262723, −14.02173449034629122572708605066, −13.43468590248873391540823018925, −12.49366175350630129516049973801, −12.04784253328219475397618218842, −11.11573632429951515876755620233, −10.424679479833169462239672057417, −10.067383091669674761533557788110, −9.25175947502119055530220579803, −8.54214727746656813335503798413, −7.15527580811011936179805331658, −6.281572688822388082143984647579, −5.5551011637934065111064308152, −4.78168108601910899570637214238, −4.11401520164730608336526311178, −3.01390196801682076246349662616, −2.93055804863845930135928774527, −1.07965129493096226739920696290, −0.38273193822596682127902602182, 1.17950640658967548073717853486, 2.32591153612227274060819574063, 3.38087953539718443343166796134, 4.1399295438677672578924877697, 5.35547710729343900352128488154, 5.83098247010273580089296197198, 6.45099871929481384809871909536, 7.19286374183336379739028809818, 7.87427039246692941887574510403, 8.7263232038788879915331248962, 9.47167914987735548663237776400, 10.442169430004186514787466477991, 11.61793809253241185310401974292, 12.16492108766243208929092129369, 12.721547678992436267251854231284, 13.47031394678597511841971360569, 14.0902516825552522730150889761, 14.83161172656699045770693543344, 15.917228749164596252623966130446, 16.24013116520660477880258422437, 17.00514791474227651390409674668, 17.6255779659202954699185676784, 18.482832750344948162194145089026, 18.99040089276889908471879124645, 19.56230435211556474422403115502

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