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Properties

Label 1-5077-5077.1035-r0-0-0
Degree $1$
Conductor $5077$
Sign $0.998 - 0.0586i$
Analytic cond. $23.5774$
Root an. cond. $23.5774$
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  + (−0.993 + 0.111i)2-s + (−0.538 − 0.842i)3-s + (0.975 − 0.220i)4-s + (0.100 + 0.994i)5-s + (0.628 + 0.777i)6-s + (−0.460 − 0.887i)7-s + (−0.944 + 0.328i)8-s + (−0.420 + 0.907i)9-s + (−0.210 − 0.977i)10-s + (−0.253 − 0.967i)11-s + (−0.711 − 0.703i)12-s + (0.593 + 0.805i)13-s + (0.556 + 0.830i)14-s + (0.784 − 0.619i)15-s + (0.902 − 0.431i)16-s + (0.951 − 0.306i)17-s + ⋯
L(s)  = 1  + (−0.993 + 0.111i)2-s + (−0.538 − 0.842i)3-s + (0.975 − 0.220i)4-s + (0.100 + 0.994i)5-s + (0.628 + 0.777i)6-s + (−0.460 − 0.887i)7-s + (−0.944 + 0.328i)8-s + (−0.420 + 0.907i)9-s + (−0.210 − 0.977i)10-s + (−0.253 − 0.967i)11-s + (−0.711 − 0.703i)12-s + (0.593 + 0.805i)13-s + (0.556 + 0.830i)14-s + (0.784 − 0.619i)15-s + (0.902 − 0.431i)16-s + (0.951 − 0.306i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5077\)
Sign: $0.998 - 0.0586i$
Analytic conductor: \(23.5774\)
Root analytic conductor: \(23.5774\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5077} (1035, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5077,\ (0:\ ),\ 0.998 - 0.0586i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9747460062 + 0.02861219967i\)
\(L(\frac12)\) \(\approx\) \(0.9747460062 + 0.02861219967i\)
\(L(1)\) \(\approx\) \(0.6515732637 - 0.07406487513i\)
\(L(1)\) \(\approx\) \(0.6515732637 - 0.07406487513i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5077 \( 1 \)
good2 \( 1 + (-0.993 + 0.111i)T \)
3 \( 1 + (-0.538 - 0.842i)T \)
5 \( 1 + (0.100 + 0.994i)T \)
7 \( 1 + (-0.460 - 0.887i)T \)
11 \( 1 + (-0.253 - 0.967i)T \)
13 \( 1 + (0.593 + 0.805i)T \)
17 \( 1 + (0.951 - 0.306i)T \)
19 \( 1 + (0.937 - 0.348i)T \)
23 \( 1 + (0.964 + 0.264i)T \)
29 \( 1 + (-0.538 + 0.842i)T \)
31 \( 1 + (0.441 - 0.897i)T \)
37 \( 1 + (0.556 + 0.830i)T \)
41 \( 1 + (0.756 - 0.654i)T \)
43 \( 1 + (0.937 + 0.348i)T \)
47 \( 1 + (0.937 + 0.348i)T \)
53 \( 1 + (-0.645 - 0.763i)T \)
59 \( 1 + (0.726 + 0.687i)T \)
61 \( 1 + (0.0556 - 0.998i)T \)
67 \( 1 + (0.996 - 0.0890i)T \)
71 \( 1 + (0.628 + 0.777i)T \)
73 \( 1 + (0.991 - 0.133i)T \)
79 \( 1 + (-0.929 + 0.369i)T \)
83 \( 1 + (-0.798 + 0.602i)T \)
89 \( 1 + (-0.645 + 0.763i)T \)
97 \( 1 + (-0.0334 + 0.999i)T \)
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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

−17.95224000057966093298724963103, −17.30084524006267280950099900542, −16.7775158407216178600976750666, −16.0332996421912563485516579483, −15.66046339293247999478849970451, −15.16864370026167200511460712328, −14.27874983751519131853296868536, −12.76198964367903519142528743559, −12.658416555306181788554966816062, −11.914519440994242243715176072110, −11.25865061882267631781676716280, −10.36999469646668825559069326098, −9.84249720490640164192536333825, −9.32672624261644085684439731492, −8.759599779102117691892980072410, −8.007834703825895180542228296675, −7.26557494617553039260335038927, −6.102746129025421544634967044622, −5.69157924405301833144840700958, −5.08887681581745295954780153611, −4.039357372926340874354595029, −3.21508573984781558505800776592, −2.4550799608881946301022829266, −1.320346207393108341063717784227, −0.60422210110365557198097320191, 0.80770377600471956529115370528, 1.18758335859948458897424337529, 2.39839508501622147744509506396, 3.03296521368276841329289619956, 3.764198426172286690829194877946, 5.27610557405225913697179515633, 5.93597843562682446813943148671, 6.56601166015840963058458254367, 7.146902914676636828630666025772, 7.59344506958013515299227505060, 8.3040697708878296732140060339, 9.362270865466967042537404082091, 9.87691405833504265761611736930, 10.91729096099250983430451253795, 11.05384436530054914585019831285, 11.617443054174400382263577493817, 12.59480278029568102331019278145, 13.46400321458484871492106971511, 14.010642148372585565359503170519, 14.54955418211236171032881932106, 15.79124849073666174400284400446, 16.12906666827418226925824710055, 16.985954829986356927580377535506, 17.23142146010091704926805098540, 18.26637023474945970061039830439

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