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Properties

Label 1-485-485.299-r0-0-0
Degree $1$
Conductor $485$
Sign $0.715 - 0.699i$
Analytic cond. $2.25233$
Root an. cond. $2.25233$
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  + (−0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s + i·4-s − 6-s + (−0.923 + 0.382i)7-s + (0.707 − 0.707i)8-s − i·9-s + (−0.707 + 0.707i)11-s + (0.707 + 0.707i)12-s + (0.923 + 0.382i)13-s + (0.923 + 0.382i)14-s − 16-s + (0.923 + 0.382i)17-s + (−0.707 + 0.707i)18-s + (0.923 + 0.382i)19-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s + i·4-s − 6-s + (−0.923 + 0.382i)7-s + (0.707 − 0.707i)8-s − i·9-s + (−0.707 + 0.707i)11-s + (0.707 + 0.707i)12-s + (0.923 + 0.382i)13-s + (0.923 + 0.382i)14-s − 16-s + (0.923 + 0.382i)17-s + (−0.707 + 0.707i)18-s + (0.923 + 0.382i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 485 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.715 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 485 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.715 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(485\)    =    \(5 \cdot 97\)
Sign: $0.715 - 0.699i$
Analytic conductor: \(2.25233\)
Root analytic conductor: \(2.25233\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{485} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 485,\ (0:\ ),\ 0.715 - 0.699i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.015174637 - 0.4137447463i\)
\(L(\frac12)\) \(\approx\) \(1.015174637 - 0.4137447463i\)
\(L(1)\) \(\approx\) \(0.8538225220 - 0.3259022846i\)
\(L(1)\) \(\approx\) \(0.8538225220 - 0.3259022846i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
97 \( 1 \)
good2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + (-0.923 + 0.382i)T \)
11 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (0.923 + 0.382i)T \)
17 \( 1 + (0.923 + 0.382i)T \)
19 \( 1 + (0.923 + 0.382i)T \)
23 \( 1 + (-0.382 + 0.923i)T \)
29 \( 1 + (0.382 - 0.923i)T \)
31 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (0.382 + 0.923i)T \)
41 \( 1 + (-0.382 + 0.923i)T \)
43 \( 1 - iT \)
47 \( 1 - iT \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (0.382 + 0.923i)T \)
61 \( 1 + T \)
67 \( 1 + (0.923 + 0.382i)T \)
71 \( 1 + (-0.382 - 0.923i)T \)
73 \( 1 + iT \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (-0.923 - 0.382i)T \)
89 \( 1 + (-0.707 + 0.707i)T \)
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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

−24.000072973814252924874098560821, −23.06305230745123108155730714592, −22.33860205159902984189633252080, −21.08906652418943456063419272865, −20.3319833054382698251781406619, −19.54596127352150217377040535641, −18.76383277881019148590251034771, −17.9730680562594136223478910608, −16.63146768405012764955774773590, −15.98967870240444692276846632382, −15.75374705805580360150699463922, −14.35484124945824907372556474245, −13.85935751423371154913621339530, −12.869226005414550008794778576162, −11.12993184766751469212647541200, −10.35179107853824217326521780210, −9.71630675003935551102327220805, −8.73030989914768851429995589226, −8.03615115187246520184627800395, −7.04100759605645803172600032527, −5.87767451182653220647785558959, −4.993256488354615543915090850759, −3.58773081120591265209366374003, −2.70262612401024369638074484982, −0.89083745436942535289915902183, 1.0887154603893283196960399781, 2.204397778431127761717055185502, 3.131960076243501933845820414182, 3.947535273738295748233925546997, 5.82058124307821963288512636487, 6.93204948087738014649121855049, 7.84179454987944147544512636130, 8.55359207291991026760867097167, 9.711177511749537646521335182475, 10.03515156610714726917979655671, 11.6855134036739424758797551709, 12.18773908697585917053921307579, 13.254254036346539574065226135505, 13.6398183600518225572704900749, 15.18545172389612063571451002473, 15.95502356314461609914709385337, 17.0190418701661441183246915823, 18.1200925460630011401417842717, 18.63711462347449812928444463895, 19.27877647889535204318590049823, 20.18434253174361177574119424907, 20.82066039981941944922080552503, 21.662640712351592241039650440892, 22.86045234048431990749233399062, 23.54003123278823322713002217286

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