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Properties

Label 1-1157-1157.7-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.673 + 0.739i$
Analytic cond. $5.37308$
Root an. cond. $5.37308$
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  + (−0.618 − 0.786i)2-s + (−0.853 − 0.520i)3-s + (−0.235 + 0.971i)4-s + (−0.415 − 0.909i)5-s + (0.118 + 0.992i)6-s + (−0.636 − 0.771i)7-s + (0.909 − 0.415i)8-s + (0.458 + 0.888i)9-s + (−0.458 + 0.888i)10-s + (−0.0950 − 0.995i)11-s + (0.707 − 0.707i)12-s + (−0.212 + 0.977i)14-s + (−0.118 + 0.992i)15-s + (−0.888 − 0.458i)16-s + (−0.618 + 0.786i)17-s + (0.415 − 0.909i)18-s + ⋯
L(s)  = 1  + (−0.618 − 0.786i)2-s + (−0.853 − 0.520i)3-s + (−0.235 + 0.971i)4-s + (−0.415 − 0.909i)5-s + (0.118 + 0.992i)6-s + (−0.636 − 0.771i)7-s + (0.909 − 0.415i)8-s + (0.458 + 0.888i)9-s + (−0.458 + 0.888i)10-s + (−0.0950 − 0.995i)11-s + (0.707 − 0.707i)12-s + (−0.212 + 0.977i)14-s + (−0.118 + 0.992i)15-s + (−0.888 − 0.458i)16-s + (−0.618 + 0.786i)17-s + (0.415 − 0.909i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.673 + 0.739i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.673 + 0.739i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1267327091 - 0.05600351563i\)
\(L(\frac12)\) \(\approx\) \(-0.1267327091 - 0.05600351563i\)
\(L(1)\) \(\approx\) \(0.2708987702 - 0.3223395518i\)
\(L(1)\) \(\approx\) \(0.2708987702 - 0.3223395518i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.618 - 0.786i)T \)
3 \( 1 + (-0.853 - 0.520i)T \)
5 \( 1 + (-0.415 - 0.909i)T \)
7 \( 1 + (-0.636 - 0.771i)T \)
11 \( 1 + (-0.0950 - 0.995i)T \)
17 \( 1 + (-0.618 + 0.786i)T \)
19 \( 1 + (0.304 - 0.952i)T \)
23 \( 1 + (-0.672 - 0.739i)T \)
29 \( 1 + (0.986 - 0.165i)T \)
31 \( 1 + (0.212 - 0.977i)T \)
37 \( 1 + (-0.965 - 0.258i)T \)
41 \( 1 + (0.0237 + 0.999i)T \)
43 \( 1 + (-0.986 - 0.165i)T \)
47 \( 1 + (-0.959 + 0.281i)T \)
53 \( 1 + (0.281 - 0.959i)T \)
59 \( 1 + (-0.520 - 0.853i)T \)
61 \( 1 + (0.560 - 0.828i)T \)
67 \( 1 + (-0.971 + 0.235i)T \)
71 \( 1 + (-0.580 - 0.814i)T \)
73 \( 1 + (0.540 + 0.841i)T \)
79 \( 1 + (-0.540 - 0.841i)T \)
83 \( 1 + (-0.800 + 0.599i)T \)
97 \( 1 + (0.0950 - 0.995i)T \)
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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

−22.28420915463178245364490002500, −21.39492101027772817636757598014, −20.19135999722218970656316483978, −19.445282089520833083864729610090, −18.50298888850725123339502136686, −18.043914993828920299962792627766, −17.47197496004490536795545740419, −16.30128822092390686883313756764, −15.80285558204984322112649286734, −15.31267981392308907021862274925, −14.565309480302067177503869173456, −13.5816920300415594063813775268, −12.16428645756189658795609238332, −11.785503875380044376934566146389, −10.57583882760531411996568707161, −10.09006658835000611434411602953, −9.41080285276031179455853235411, −8.41420174419778167814050060963, −7.20923282319536079711538658640, −6.77281734543008285672015370519, −5.91731085099758374323402315486, −5.13407910470224696264272867669, −4.169866962157511967095089312863, −2.99973490361496009507428265511, −1.67162695511409573983772362707, 0.11188952464355537707353481301, 0.79853659075226424405799780137, 1.83977946769252961409188188584, 3.14641728247388952343402736137, 4.173196843098280777322577281614, 4.8697697147020436586237081070, 6.18835325361928551707877449599, 6.95580489965686629202469718738, 8.03316800582303203185916446047, 8.500804890559081187961614492399, 9.64703063677359786987225832855, 10.44408061010493044759345835735, 11.23502374984863111187556032411, 11.79878283782356308289018847737, 12.76503191106320087358149543721, 13.1820204624837530124601124860, 13.848221013114947185562118685854, 15.72134424252218660774738098229, 16.264600408230683579774149324481, 16.90414560395982897805453322195, 17.47862144820622223789138330934, 18.3223157764430161019444144264, 19.34909092486977234587977024432, 19.56450405463532242758035983799, 20.40741130910483091475266435099

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