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Properties

Label 1-5200-5200.1403-r0-0-0
Degree $1$
Conductor $5200$
Sign $0.171 - 0.985i$
Analytic cond. $24.1486$
Root an. cond. $24.1486$
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  + (−0.309 − 0.951i)3-s + i·7-s + (−0.809 + 0.587i)9-s + (0.587 − 0.809i)11-s + (−0.951 − 0.309i)17-s + (0.951 + 0.309i)19-s + (0.951 − 0.309i)21-s + (0.587 − 0.809i)23-s + (0.809 + 0.587i)27-s + (−0.951 + 0.309i)29-s + (0.309 − 0.951i)31-s + (−0.951 − 0.309i)33-s + (0.809 − 0.587i)37-s + (−0.809 + 0.587i)41-s + 43-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)3-s + i·7-s + (−0.809 + 0.587i)9-s + (0.587 − 0.809i)11-s + (−0.951 − 0.309i)17-s + (0.951 + 0.309i)19-s + (0.951 − 0.309i)21-s + (0.587 − 0.809i)23-s + (0.809 + 0.587i)27-s + (−0.951 + 0.309i)29-s + (0.309 − 0.951i)31-s + (−0.951 − 0.309i)33-s + (0.809 − 0.587i)37-s + (−0.809 + 0.587i)41-s + 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5200\)    =    \(2^{4} \cdot 5^{2} \cdot 13\)
Sign: $0.171 - 0.985i$
Analytic conductor: \(24.1486\)
Root analytic conductor: \(24.1486\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5200} (1403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5200,\ (0:\ ),\ 0.171 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.084320127 - 0.9114662908i\)
\(L(\frac12)\) \(\approx\) \(1.084320127 - 0.9114662908i\)
\(L(1)\) \(\approx\) \(0.9266539101 - 0.2762849205i\)
\(L(1)\) \(\approx\) \(0.9266539101 - 0.2762849205i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 + iT \)
11 \( 1 + (0.587 - 0.809i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (0.951 + 0.309i)T \)
23 \( 1 + (0.587 - 0.809i)T \)
29 \( 1 + (-0.951 + 0.309i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.587 + 0.809i)T \)
61 \( 1 + (-0.587 + 0.809i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (-0.951 + 0.309i)T \)
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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

−17.711317722734508236450507564662, −17.424862638387929231206481224389, −16.9028766358075835997839339650, −16.12514144044137854752654283572, −15.440358798268094759591012604516, −15.02305996950096509542397739566, −14.0985454539679926002297831734, −13.67712910430814553784210567552, −12.75498785212593509268086365807, −11.99390196969683835450064245216, −11.19210272448547758781773698734, −10.85865557088755334477233851703, −9.98546208778638344969338086177, −9.45812466559080075971065183122, −8.92804848289534879524425351921, −7.879057982972850964660018967603, −7.12073828802331695452845882881, −6.55360429687299300979113352071, −5.624949792929021459671465282310, −4.83839940241818871751472854756, −4.27978590806753608166899786080, −3.65775138928796752274648374537, −2.88831454963978541408051083634, −1.72048756624885263657564102580, −0.80709738416716340686991352062, 0.51363363035261398225418477908, 1.39526355011439898950603786107, 2.335089026172164346414606613073, 2.82763535156031057081162356552, 3.83831087918525277995972936993, 4.88489654013041985160947443894, 5.64192851781562288116873523154, 6.123481181647722290482125669452, 6.8224789695500438486388855257, 7.58270204069638209225597484780, 8.33351432245104628879213412850, 8.985046968527971585938225419989, 9.47867126509804990562976442660, 10.74679059261157832184485655551, 11.27406027053330960786879445851, 11.87710803086601023297380577404, 12.36674447026004556751744542872, 13.32356844099874438184838390507, 13.54569618332396555036141364727, 14.61409730104291882291084477617, 14.95925446183194393549870502465, 16.10457419306937811539815654062, 16.44674911904648751683108552023, 17.28185413162323280927543967931, 17.958361047651556634165122306590

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