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Properties

Label 1-1148-1148.319-r1-0-0
Degree $1$
Conductor $1148$
Sign $-0.390 - 0.920i$
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  + (0.866 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.390 - 0.920i$
Analytic conductor: \(123.369\)
Root analytic conductor: \(123.369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (1:\ ),\ -0.390 - 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.363347617 - 3.567992180i\)
\(L(\frac12)\) \(\approx\) \(2.363347617 - 3.567992180i\)
\(L(1)\) \(\approx\) \(1.622896919 - 0.8889758622i\)
\(L(1)\) \(\approx\) \(1.622896919 - 0.8889758622i\)

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

−21.28616211123232094725690182333, −20.74837125401036653652550477416, −19.63058209135607432750446924446, −19.230629618122802918262430234309, −18.408353156564147681254764796385, −17.45817391220033351379348896305, −16.71591467828127569066604001212, −15.75135280384889146669289408215, −14.98322528120486001706781949880, −14.264128604086014624921354677325, −13.9272944894253063887816634296, −12.91400668993220412664100803230, −11.78957815597759899893059372099, −10.9919777837635737070447941009, −10.06760717299876848478243456946, −9.42860089490360797715679587820, −8.87711673057987577574751511865, −7.51930389791765529330789488775, −7.09013230375332813560698889266, −5.990162470981377943220188979028, −4.94098786825126910559165614974, −3.86857426815689117179177506512, −3.244576423151170104068500579000, −2.184314688789911851603493054493, −1.39679811196676239119269664493, 0.84750725468168894941809450963, 1.21997393412780041161130242870, 2.56654874826957497976972198056, 3.34010756536974493526081186924, 4.3930251324592755288614002382, 5.49428897930384600529521408721, 6.255175453171449919639890097407, 7.34182199330929328841507327463, 8.1783411099671951995614874747, 8.79911061537021960624353570938, 9.61066579079858422454868900968, 10.28042326318231536014283771652, 11.721535546037492443109242280410, 12.37028076157404221167715468380, 13.044723055291335079913765857390, 14.000323187141140991997583528635, 14.26243802336142146232839274892, 15.43141565046932251145895451537, 16.18846276663665719867097970701, 17.12048494214651579652841817603, 17.74054233765731402609592625273, 18.73176982359000875245491203272, 19.29085740776783365756746551835, 20.38891725342526682444224578524, 20.52854151380092498527201865785

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