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Properties

Label 1-980-980.919-r1-0-0
Degree $1$
Conductor $980$
Sign $-0.860 + 0.509i$
Analytic cond. $105.315$
Root an. cond. $105.315$
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  + (0.0747 − 0.997i)3-s + (−0.988 − 0.149i)9-s + (0.988 − 0.149i)11-s + (−0.623 − 0.781i)13-s + (−0.955 − 0.294i)17-s + (0.5 + 0.866i)19-s + (0.955 − 0.294i)23-s + (−0.222 + 0.974i)27-s + (−0.222 − 0.974i)29-s + (0.5 − 0.866i)31-s + (−0.0747 − 0.997i)33-s + (0.733 − 0.680i)37-s + (−0.826 + 0.563i)39-s + (−0.900 + 0.433i)41-s + (−0.900 − 0.433i)43-s + ⋯
L(s)  = 1  + (0.0747 − 0.997i)3-s + (−0.988 − 0.149i)9-s + (0.988 − 0.149i)11-s + (−0.623 − 0.781i)13-s + (−0.955 − 0.294i)17-s + (0.5 + 0.866i)19-s + (0.955 − 0.294i)23-s + (−0.222 + 0.974i)27-s + (−0.222 − 0.974i)29-s + (0.5 − 0.866i)31-s + (−0.0747 − 0.997i)33-s + (0.733 − 0.680i)37-s + (−0.826 + 0.563i)39-s + (−0.900 + 0.433i)41-s + (−0.900 − 0.433i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.860 + 0.509i$
Analytic conductor: \(105.315\)
Root analytic conductor: \(105.315\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (919, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 980,\ (1:\ ),\ -0.860 + 0.509i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2209166916 - 0.8071965184i\)
\(L(\frac12)\) \(\approx\) \(-0.2209166916 - 0.8071965184i\)
\(L(1)\) \(\approx\) \(0.8459285258 - 0.4396425879i\)
\(L(1)\) \(\approx\) \(0.8459285258 - 0.4396425879i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.0747 - 0.997i)T \)
11 \( 1 + (0.988 - 0.149i)T \)
13 \( 1 + (-0.623 - 0.781i)T \)
17 \( 1 + (-0.955 - 0.294i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.955 - 0.294i)T \)
29 \( 1 + (-0.222 - 0.974i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.733 - 0.680i)T \)
41 \( 1 + (-0.900 + 0.433i)T \)
43 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (0.365 - 0.930i)T \)
53 \( 1 + (0.733 + 0.680i)T \)
59 \( 1 + (-0.826 + 0.563i)T \)
61 \( 1 + (-0.733 + 0.680i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.222 - 0.974i)T \)
73 \( 1 + (-0.365 - 0.930i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (-0.988 - 0.149i)T \)
97 \( 1 - T \)
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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

−21.869606781613664830155082178316, −21.44894477062776483406614981650, −20.26035061818576127218253247483, −19.84735509444516419130607110944, −19.06099989712813285454892308283, −17.82542491169287736593988503110, −17.11031200106796872293920588494, −16.53228012266723763138236167522, −15.56349171971970108424847565126, −14.94716934528180011934479580286, −14.19178761118407299878489855000, −13.40318974257011272879059703233, −12.197186091468485892987667240845, −11.41326746365744466092502444967, −10.793245053139062161548826464818, −9.660745458629755045339208339133, −9.17732661579324583594491427859, −8.43919401368963813628060279590, −7.06687422802185437581557690477, −6.43117990916939753203297527147, −5.07261740397288102812473936600, −4.5854281386293959737809232748, −3.578988887418509042391765199997, −2.666803998727235743467338552457, −1.39978988518887253969463952495, 0.18089084347681942642795310501, 1.15591029875110653462210709347, 2.2412210131329778134948834084, 3.127187413309015845177641072144, 4.284713758108889235639716362689, 5.48329174877790071503977986749, 6.28610086316406737863476120066, 7.11840669586261721520156617106, 7.87230739311954440300095195421, 8.76425520507040228294299581608, 9.576290474803849156171268529041, 10.69123860443183221106979817561, 11.72187299557193060946742147739, 12.12533174501327450947126828519, 13.228707802235611186356464124347, 13.671108924479339676928363014969, 14.75151928118693813894705363146, 15.23982984073496872056546851611, 16.70332650988781195194536654336, 17.085825903708824208391964382160, 18.03998617244786707101728548015, 18.64963936465567460590645116961, 19.57561664226252224187274455940, 20.04345246359939600640689636152, 20.87769427092579030616457777192

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