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Properties

Label 1-1148-1148.255-r0-0-0
Degree $1$
Conductor $1148$
Sign $0.834 - 0.550i$
Analytic cond. $5.33128$
Root an. cond. $5.33128$
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  + (0.866 − 0.5i)3-s + (−0.5 + 0.866i)5-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)11-s − i·13-s − i·15-s + (0.866 − 0.5i)17-s + (0.866 + 0.5i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − i·27-s + i·29-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)33-s + (−0.5 + 0.866i)37-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (−0.5 + 0.866i)5-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)11-s − i·13-s − i·15-s + (0.866 − 0.5i)17-s + (0.866 + 0.5i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s − i·27-s + i·29-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)33-s + (−0.5 + 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.834 - 0.550i$
Analytic conductor: \(5.33128\)
Root analytic conductor: \(5.33128\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (0:\ ),\ 0.834 - 0.550i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.788454003 - 0.5366951687i\)
\(L(\frac12)\) \(\approx\) \(1.788454003 - 0.5366951687i\)
\(L(1)\) \(\approx\) \(1.311184378 - 0.1623766118i\)
\(L(1)\) \(\approx\) \(1.311184378 - 0.1623766118i\)

Euler product

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

−21.31719859199028389796313722592, −20.66534863803082077742443668610, −19.83471260461104302249716499150, −19.207918146337312220668721206472, −18.603229962520060807635750560587, −17.32652823802890752669198677843, −16.50280560202210163143981189172, −15.865960708338543382400536432932, −15.37398959695045015192595541556, −14.27314128206291270892359889496, −13.6417785391555697709750366624, −12.89475186916725969601586483036, −11.96714062501419888700693657810, −11.08923717141742186387234261924, −10.167367300861780641576841073, −9.20382990757319620329506621237, −8.78157539399408464767472801929, −7.769000929621166163306266100366, −7.30656558497701941091428108339, −5.66311669589960431184279942442, −5.00182050317199751000393482937, −4.02087606413931808443717810157, −3.34963328232085958701755498656, −2.23856601981751769269283830479, −1.08895752058571370521737213276, 0.81614916179104739918919884515, 2.21685984247121371154384209216, 3.02071955602975435780776353319, 3.56036455711411102059501916614, 4.869331747544792329790716975371, 5.93263058733235471613335824535, 7.05368225724602782511308658280, 7.62505718315334496773091647845, 8.12827079886756880441803190133, 9.294566399591163794157274192974, 10.17625020983490018943603530298, 10.80117394332482664473053057779, 12.04740027216762416563420500889, 12.563655386593644246777829736642, 13.50366323878876246636722194446, 14.2936011741147267827496292817, 14.9647070376339083379138750104, 15.53444876319289372789376977460, 16.42405525798944815909252238564, 17.73696460288670478914093111754, 18.39657708678386439741118105753, 18.746176076228633325335590773887, 19.68051298994715335306489656598, 20.548764862531794678623262536230, 20.80212883406785739427475423852

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