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Properties

Label 1-979-979.791-r1-0-0
Degree $1$
Conductor $979$
Sign $0.960 - 0.279i$
Analytic cond. $105.208$
Root an. cond. $105.208$
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  + (0.654 + 0.755i)2-s + (−0.989 + 0.142i)3-s + (−0.142 + 0.989i)4-s + (−0.841 + 0.540i)5-s + (−0.755 − 0.654i)6-s + (0.540 + 0.841i)7-s + (−0.841 + 0.540i)8-s + (0.959 − 0.281i)9-s + (−0.959 − 0.281i)10-s − i·12-s + (−0.989 + 0.142i)13-s + (−0.281 + 0.959i)14-s + (0.755 − 0.654i)15-s + (−0.959 − 0.281i)16-s + (−0.654 + 0.755i)17-s + (0.841 + 0.540i)18-s + ⋯
L(s)  = 1  + (0.654 + 0.755i)2-s + (−0.989 + 0.142i)3-s + (−0.142 + 0.989i)4-s + (−0.841 + 0.540i)5-s + (−0.755 − 0.654i)6-s + (0.540 + 0.841i)7-s + (−0.841 + 0.540i)8-s + (0.959 − 0.281i)9-s + (−0.959 − 0.281i)10-s − i·12-s + (−0.989 + 0.142i)13-s + (−0.281 + 0.959i)14-s + (0.755 − 0.654i)15-s + (−0.959 − 0.281i)16-s + (−0.654 + 0.755i)17-s + (0.841 + 0.540i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 979 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 979 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(979\)    =    \(11 \cdot 89\)
Sign: $0.960 - 0.279i$
Analytic conductor: \(105.208\)
Root analytic conductor: \(105.208\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{979} (791, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 979,\ (1:\ ),\ 0.960 - 0.279i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2161579590 + 0.03085951384i\)
\(L(\frac12)\) \(\approx\) \(-0.2161579590 + 0.03085951384i\)
\(L(1)\) \(\approx\) \(0.4788201553 + 0.5974324382i\)
\(L(1)\) \(\approx\) \(0.4788201553 + 0.5974324382i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.654 + 0.755i)T \)
3 \( 1 + (-0.989 + 0.142i)T \)
5 \( 1 + (-0.841 + 0.540i)T \)
7 \( 1 + (0.540 + 0.841i)T \)
13 \( 1 + (-0.989 + 0.142i)T \)
17 \( 1 + (-0.654 + 0.755i)T \)
19 \( 1 + (0.281 + 0.959i)T \)
23 \( 1 + (0.281 + 0.959i)T \)
29 \( 1 + (-0.540 - 0.841i)T \)
31 \( 1 + (0.281 - 0.959i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.989 - 0.142i)T \)
43 \( 1 + (-0.540 + 0.841i)T \)
47 \( 1 + (0.142 - 0.989i)T \)
53 \( 1 + (0.142 + 0.989i)T \)
59 \( 1 + (-0.989 - 0.142i)T \)
61 \( 1 + (-0.909 + 0.415i)T \)
67 \( 1 + (-0.142 - 0.989i)T \)
71 \( 1 + (-0.841 - 0.540i)T \)
73 \( 1 + (0.959 + 0.281i)T \)
79 \( 1 + (-0.959 - 0.281i)T \)
83 \( 1 + (-0.755 - 0.654i)T \)
97 \( 1 + (0.841 - 0.540i)T \)
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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

−20.70433224655171611010739578204, −20.064774631740207702827881360653, −19.47038556061083576541184562393, −18.45575948866526814939865675974, −17.69196283227024623911511136386, −16.843991294776358517339959061526, −16.00426393796064454736262673252, −15.22714700931026225245801044281, −14.27269062426737255140981725351, −13.33071459185100556397593024164, −12.59461804751769508770700256220, −11.95371137159667291670787563755, −11.17187624057247409884989773666, −10.717608832215147494767633393015, −9.70658676941135972851392621027, −8.64784365724746326073661909518, −7.23535634601192905612985652137, −6.89074626988529379228122194605, −5.36150530265904647256337990808, −4.76466515098613916929241884637, −4.3135920279804161860880515528, −3.07228320696648016985906698853, −1.72366384549861239803180533058, −0.6816799580852268730088949047, −0.06233141581178334281519120634, 1.90048801315086586046564215604, 3.19256827408069467137327196931, 4.21144848282303037756787993577, 4.87518224749004277672112271932, 5.809318604751653350657712593947, 6.47561397075137434471238316965, 7.501122459656511375850875495780, 8.02256319287447103156196771076, 9.22833985605584707678893803514, 10.34853204388376206040849771695, 11.59336999200051219529764734868, 11.7194506133663900122191195707, 12.54475658436069776933998684051, 13.53357106807656797883796706645, 14.76096556440358612225117803343, 15.21789055516324098381534326320, 15.68366526857735952692782753296, 16.86404449322850988295476295826, 17.21073801858526910169375980122, 18.32049803625411922211789215803, 18.75183737730881965330123394544, 19.98901489750432208219352797913, 21.17302150060271859341924637721, 21.75922215308167845024446874191, 22.397226827791676177241929873763

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