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Properties

Label 1-1148-1148.719-r1-0-0
Degree $1$
Conductor $1148$
Sign $-0.507 + 0.861i$
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  + (0.258 + 0.965i)3-s + (0.743 + 0.669i)5-s + (−0.866 + 0.5i)9-s + (0.998 + 0.0523i)11-s + (0.987 − 0.156i)13-s + (−0.453 + 0.891i)15-s + (0.0523 − 0.998i)17-s + (0.358 + 0.933i)19-s + (−0.104 + 0.994i)23-s + (0.104 + 0.994i)25-s + (−0.707 − 0.707i)27-s + (0.891 + 0.453i)29-s + (−0.669 − 0.743i)31-s + (0.207 + 0.978i)33-s + (0.669 − 0.743i)37-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)3-s + (0.743 + 0.669i)5-s + (−0.866 + 0.5i)9-s + (0.998 + 0.0523i)11-s + (0.987 − 0.156i)13-s + (−0.453 + 0.891i)15-s + (0.0523 − 0.998i)17-s + (0.358 + 0.933i)19-s + (−0.104 + 0.994i)23-s + (0.104 + 0.994i)25-s + (−0.707 − 0.707i)27-s + (0.891 + 0.453i)29-s + (−0.669 − 0.743i)31-s + (0.207 + 0.978i)33-s + (0.669 − 0.743i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.507 + 0.861i$
Analytic conductor: \(123.369\)
Root analytic conductor: \(123.369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (1:\ ),\ -0.507 + 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.597097272 + 2.796016529i\)
\(L(\frac12)\) \(\approx\) \(1.597097272 + 2.796016529i\)
\(L(1)\) \(\approx\) \(1.300186876 + 0.7572156456i\)
\(L(1)\) \(\approx\) \(1.300186876 + 0.7572156456i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (0.743 + 0.669i)T \)
11 \( 1 + (0.998 + 0.0523i)T \)
13 \( 1 + (0.987 - 0.156i)T \)
17 \( 1 + (0.0523 - 0.998i)T \)
19 \( 1 + (0.358 + 0.933i)T \)
23 \( 1 + (-0.104 + 0.994i)T \)
29 \( 1 + (0.891 + 0.453i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (0.669 - 0.743i)T \)
43 \( 1 + (-0.587 + 0.809i)T \)
47 \( 1 + (-0.777 + 0.629i)T \)
53 \( 1 + (0.838 - 0.544i)T \)
59 \( 1 + (0.913 - 0.406i)T \)
61 \( 1 + (0.406 - 0.913i)T \)
67 \( 1 + (0.544 + 0.838i)T \)
71 \( 1 + (0.453 + 0.891i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.965 - 0.258i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.933 + 0.358i)T \)
97 \( 1 + (-0.453 + 0.891i)T \)
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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

−20.7502828379332920473312630005, −19.92249534422810665593453550283, −19.48812801884241103270834126175, −18.35563583134529209216127208779, −17.90435063867756309854938362084, −17.00641473781925763383842294223, −16.51387171234424573545654829687, −15.28809151520428524613971333464, −14.37532076509657648309207423053, −13.68798677898171532536722693771, −13.123516836317572952576911655011, −12.314593128229677857461522602957, −11.61691544909331617405741100142, −10.58337044252018851838033116567, −9.50729246133946068896375192392, −8.61109775949438730865146045559, −8.37882343888598846659422195200, −6.91158876702686597494221290711, −6.3872893614121504426037094516, −5.616472191383291434796211905582, −4.448488224492915193947553250743, −3.39093097260808044913054385546, −2.230478916533770287386539846969, −1.39779186283012917837440410627, −0.676958477730275737843077490736, 1.1602699886507236860189177919, 2.30198872663138426088551661989, 3.35836069103483288979449546848, 3.887964804305973184537486401013, 5.15626824383157748051680795874, 5.87430386153361093429468314419, 6.72250048952768854982654656951, 7.830317583776557095381193832007, 8.83154576513920629560712894919, 9.64424257871358808482238022148, 10.02162221916559570985879142378, 11.225645045536192203324066314474, 11.47721812331251770986092014133, 12.92210242696726542887485598957, 13.916800777425429883133439944500, 14.29164219535215992584971798277, 15.04983173433228084015792661350, 16.03027205217195661516903015618, 16.5459048569680461200756770259, 17.55557561330446594057822363496, 18.16821753506091944062167763942, 19.090981766430093929957952701969, 20.0091428668138614974748682369, 20.67046677080889039738122581997, 21.43280677587218636380781629407

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