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Properties

Label 1-812-812.391-r1-0-0
Degree $1$
Conductor $812$
Sign $0.965 + 0.259i$
Analytic cond. $87.2615$
Root an. cond. $87.2615$
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)3-s + (−0.222 − 0.974i)5-s + (−0.623 − 0.781i)9-s + (0.781 + 0.623i)11-s + (0.623 − 0.781i)13-s + (0.974 + 0.222i)15-s + i·17-s + (0.433 + 0.900i)19-s + (0.222 − 0.974i)23-s + (−0.900 + 0.433i)25-s + (0.974 − 0.222i)27-s + (−0.974 + 0.222i)31-s + (−0.900 + 0.433i)33-s + (−0.781 + 0.623i)37-s + (0.433 + 0.900i)39-s + ⋯
L(s)  = 1  + (−0.433 + 0.900i)3-s + (−0.222 − 0.974i)5-s + (−0.623 − 0.781i)9-s + (0.781 + 0.623i)11-s + (0.623 − 0.781i)13-s + (0.974 + 0.222i)15-s + i·17-s + (0.433 + 0.900i)19-s + (0.222 − 0.974i)23-s + (−0.900 + 0.433i)25-s + (0.974 − 0.222i)27-s + (−0.974 + 0.222i)31-s + (−0.900 + 0.433i)33-s + (−0.781 + 0.623i)37-s + (0.433 + 0.900i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(812\)    =    \(2^{2} \cdot 7 \cdot 29\)
Sign: $0.965 + 0.259i$
Analytic conductor: \(87.2615\)
Root analytic conductor: \(87.2615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{812} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 812,\ (1:\ ),\ 0.965 + 0.259i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.630135477 + 0.2154626732i\)
\(L(\frac12)\) \(\approx\) \(1.630135477 + 0.2154626732i\)
\(L(1)\) \(\approx\) \(0.9517013155 + 0.1125079299i\)
\(L(1)\) \(\approx\) \(0.9517013155 + 0.1125079299i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
29 \( 1 \)
good3 \( 1 + (-0.433 + 0.900i)T \)
5 \( 1 + (-0.222 - 0.974i)T \)
11 \( 1 + (0.781 + 0.623i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
17 \( 1 + iT \)
19 \( 1 + (0.433 + 0.900i)T \)
23 \( 1 + (0.222 - 0.974i)T \)
31 \( 1 + (-0.974 + 0.222i)T \)
37 \( 1 + (-0.781 + 0.623i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.974 - 0.222i)T \)
47 \( 1 + (0.781 + 0.623i)T \)
53 \( 1 + (-0.222 - 0.974i)T \)
59 \( 1 + T \)
61 \( 1 + (0.433 - 0.900i)T \)
67 \( 1 + (0.623 + 0.781i)T \)
71 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (0.974 + 0.222i)T \)
79 \( 1 + (-0.781 + 0.623i)T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (0.974 - 0.222i)T \)
97 \( 1 + (0.433 + 0.900i)T \)
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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

−22.06373270275110158215859425656, −21.50768423163882883011852780701, −20.085418132290635607191237349684, −19.45700869126552828798916148233, −18.65295085115692363420027029819, −18.19723910011048586381359105988, −17.30631796676732836276863951771, −16.42483939510701565516799618612, −15.61465102541750096889534751825, −14.436461793834737673431615218808, −13.8401106721862322508649432871, −13.200446837023743422030781150120, −11.8065315575150958921966069630, −11.49173918967544873029126655650, −10.82903393262953512803218288949, −9.49981341588200535103852218835, −8.63052848714155998272674339019, −7.4220283957171582704277513750, −6.93669312462765441070534037066, −6.15435656219510719410677503908, −5.20191614023857814348396612416, −3.811739396806747443364326330948, −2.895546728958327065127181332202, −1.78824100785608635936829710988, −0.66277461938204213484040665615, 0.640761363806244289401308861897, 1.737459500603156902096997814080, 3.51416723036408021611930144244, 4.030442097774613574119199487224, 5.07063064338541377597921173049, 5.74945148099635242897293173170, 6.779812453157478333523251943283, 8.19610417345402361098239993320, 8.753730355483468377529513756114, 9.72128118014095509238769344812, 10.44255049226571541503059308743, 11.37405786684351988856078589735, 12.32407242843906342773967149447, 12.76112609130609427775531299476, 14.10547061196898733580185734609, 14.9639395124007628628815807153, 15.67614354128674981451963602598, 16.467020243205754984008123161215, 17.100404999514990534994772822278, 17.748837843124435759106281932461, 18.89950215808269555013194782935, 20.066479488649052735621867963456, 20.44505138894909160048816166017, 21.1369265451978418027215415112, 22.157576563497201904882402886895

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