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Properties

Label 1-1148-1148.495-r1-0-0
Degree $1$
Conductor $1148$
Sign $0.998 + 0.0598i$
Analytic cond. $123.369$
Root an. cond. $123.369$
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  + (0.965 − 0.258i)3-s + (−0.866 + 0.5i)5-s + (0.866 − 0.5i)9-s + (0.965 − 0.258i)11-s + (0.707 + 0.707i)13-s + (−0.707 + 0.707i)15-s + (0.258 + 0.965i)17-s + (0.965 + 0.258i)19-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s + (0.707 − 0.707i)27-s + (−0.707 − 0.707i)29-s + (0.5 − 0.866i)31-s + (0.866 − 0.5i)33-s + (−0.5 − 0.866i)37-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)3-s + (−0.866 + 0.5i)5-s + (0.866 − 0.5i)9-s + (0.965 − 0.258i)11-s + (0.707 + 0.707i)13-s + (−0.707 + 0.707i)15-s + (0.258 + 0.965i)17-s + (0.965 + 0.258i)19-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s + (0.707 − 0.707i)27-s + (−0.707 − 0.707i)29-s + (0.5 − 0.866i)31-s + (0.866 − 0.5i)33-s + (−0.5 − 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.998 + 0.0598i$
Analytic conductor: \(123.369\)
Root analytic conductor: \(123.369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (495, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (1:\ ),\ 0.998 + 0.0598i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.303056719 + 0.09899928170i\)
\(L(\frac12)\) \(\approx\) \(3.303056719 + 0.09899928170i\)
\(L(1)\) \(\approx\) \(1.535144523 + 0.004893056733i\)
\(L(1)\) \(\approx\) \(1.535144523 + 0.004893056733i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good3 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (0.965 - 0.258i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 + (0.258 + 0.965i)T \)
19 \( 1 + (0.965 + 0.258i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.965 + 0.258i)T \)
53 \( 1 + (-0.965 + 0.258i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (0.258 + 0.965i)T \)
71 \( 1 + (0.707 + 0.707i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (-0.258 + 0.965i)T \)
83 \( 1 + T \)
89 \( 1 + (0.258 - 0.965i)T \)
97 \( 1 + (-0.707 + 0.707i)T \)
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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

−20.64362439550677958964681237745, −20.40244098274751923497519828413, −19.73996383086424261452372609340, −18.974630896298820371194761498221, −18.17936652805954762699847579107, −17.162017990279349896275568539092, −16.09774972903081573386055079198, −15.73744739628526492106829949918, −14.97511912593847076391075909145, −14.02586201928270495437023251834, −13.47713472246928467841372180106, −12.41730169006536283860283174424, −11.77068144244201710617993395913, −10.83229878144080669782492014036, −9.73659149530323163685289617474, −9.10245384162870753058084739073, −8.36092657578422340251366494170, −7.556698589651971381701786368527, −6.89885919850915967978302724012, −5.40785433230892764854843191252, −4.627841956750083734794495833185, −3.53751457158921214273038565555, −3.24592373991085471556452052409, −1.7034183594488819986267109645, −0.81571186200902368759803747544, 0.83457817186666330763388687087, 1.853056371650914722028882980323, 2.95561309115026959320087068377, 3.93228333342514991025453307455, 4.15802705944936204151615522787, 5.98569862215314949978275615933, 6.684521442720796669679514606870, 7.609120671459455344788986302169, 8.24518123993337313693926804146, 9.03708652456711995581457211022, 9.87426974859159183507279216363, 10.92704030727811514060313946844, 11.74568308165624695407128835387, 12.43113342485882062380129239444, 13.453620390336074192706665719956, 14.30521600240065009358907294359, 14.658138918233250678791168536083, 15.64065974830327304161325592008, 16.245030840227328734211108973936, 17.2728290138038515772053463602, 18.41695314014864506674859805951, 18.90640221803137979781026859485, 19.47661353809121837811280254961, 20.256108659203061229935000260951, 20.9254969115203277177974085903

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