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Properties

Label 1-1148-1148.79-r0-0-0
Degree $1$
Conductor $1148$
Sign $0.952 - 0.304i$
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  + (0.965 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.952 - 0.304i$
Analytic conductor: \(5.33128\)
Root analytic conductor: \(5.33128\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (0:\ ),\ 0.952 - 0.304i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.652348237 - 0.4137992321i\)
\(L(\frac12)\) \(\approx\) \(2.652348237 - 0.4137992321i\)
\(L(1)\) \(\approx\) \(1.724918348 - 0.1954067694i\)
\(L(1)\) \(\approx\) \(1.724918348 - 0.1954067694i\)

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

−21.15922514111326292550059980471, −20.652254565028897851278633943196, −20.001749188080367142729936003651, −18.85507059689958030342050735476, −18.367361959048531770764394279831, −17.728326570386245963148593858000, −16.56914322790737823916164107201, −15.6096073288354791854436807260, −15.30154679725975475305052517318, −14.075037038954057570813106251612, −13.62188316699477564594727964877, −13.18738991460592141700634043202, −11.8873404023801407328111589881, −10.81371065508267618232051128367, −10.11056914822520852856089755384, −9.53392287853886197781588327373, −8.60423219182702289304503093178, −7.71978018101096980952913185976, −7.06079954214107724692431593124, −5.74371518383549827381958657312, −5.20658434440475929912015670562, −3.85408884512546682053357746262, −2.94276606702407043910118167111, −2.42864188630339808203906800499, −1.176217790018533428455467925389, 1.236021967610240754749594384541, 1.95669983225392066294873374893, 2.89991052892708767097690089613, 3.921830827934725112821958134007, 4.91886022803364982272131283379, 5.90401540870254720588689516765, 6.78251253312226315016664573963, 7.78735323326525546876838844057, 8.57287116716312925936688460299, 9.174458421636391407998695697795, 10.10945822816860947333733326032, 10.69661139352927625035924135589, 12.244476618883349952053330698713, 12.69104765049001851916715083883, 13.58487589135595684048102115157, 14.09505260457604834451008567674, 14.846443266319960186736802004892, 15.999846676036115535645171637652, 16.359589612662304679699917888214, 17.68512259303003784720810080545, 18.19017512817339742457710831764, 18.88343243662948982032316847354, 19.91785996862098087156694446084, 20.45597740032715527881465982248, 21.33259282694497856548941210538

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