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Properties

Label 1-1275-1275.548-r0-0-0
Degree $1$
Conductor $1275$
Sign $0.456 - 0.889i$
Analytic cond. $5.92107$
Root an. cond. $5.92107$
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  + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + 7-s + (−0.587 − 0.809i)8-s + (−0.951 − 0.309i)11-s + (−0.951 + 0.309i)13-s + (−0.951 − 0.309i)14-s + (0.309 + 0.951i)16-s + (−0.809 + 0.587i)19-s + (0.809 + 0.587i)22-s + (0.309 − 0.951i)23-s + 26-s + (0.809 + 0.587i)28-s + (−0.587 + 0.809i)29-s + (0.587 + 0.809i)31-s − i·32-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + 7-s + (−0.587 − 0.809i)8-s + (−0.951 − 0.309i)11-s + (−0.951 + 0.309i)13-s + (−0.951 − 0.309i)14-s + (0.309 + 0.951i)16-s + (−0.809 + 0.587i)19-s + (0.809 + 0.587i)22-s + (0.309 − 0.951i)23-s + 26-s + (0.809 + 0.587i)28-s + (−0.587 + 0.809i)29-s + (0.587 + 0.809i)31-s − i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1275\)    =    \(3 \cdot 5^{2} \cdot 17\)
Sign: $0.456 - 0.889i$
Analytic conductor: \(5.92107\)
Root analytic conductor: \(5.92107\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1275} (548, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1275,\ (0:\ ),\ 0.456 - 0.889i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7164354459 - 0.4374695914i\)
\(L(\frac12)\) \(\approx\) \(0.7164354459 - 0.4374695914i\)
\(L(1)\) \(\approx\) \(0.6935137244 - 0.1316379236i\)
\(L(1)\) \(\approx\) \(0.6935137244 - 0.1316379236i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.951 - 0.309i)T \)
7 \( 1 + T \)
11 \( 1 + (-0.951 - 0.309i)T \)
13 \( 1 + (-0.951 + 0.309i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (-0.587 + 0.809i)T \)
31 \( 1 + (0.587 + 0.809i)T \)
37 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (0.951 - 0.309i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (0.587 - 0.809i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (0.951 + 0.309i)T \)
67 \( 1 + (0.587 + 0.809i)T \)
71 \( 1 + (0.587 - 0.809i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.587 + 0.809i)T \)
83 \( 1 + (-0.587 - 0.809i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−21.01672294940518277295284309751, −20.32945078269522701225856067028, −19.51243619899693615488341796508, −18.79412311663233407532455669911, −18.00885811360381589177673329878, −17.308942170026109582898393181786, −16.96268119193815877909603201203, −15.647740595120135661874670523891, −15.23776343938103086260248878269, −14.56269322931734989074098724176, −13.52160708252036176277568039893, −12.53459038009686498262502553063, −11.52100947456967690720096270124, −10.98710712255097957428970075443, −10.08748035594624981194773301866, −9.43429861760098481379025994252, −8.39030046093581212311240699815, −7.746440377692739929951110284744, −7.21732627163290069112290930398, −6.022563545502461977842020431355, −5.20136439978645723136437888979, −4.41747010182479996227767947755, −2.71931071792627233893826301417, −2.14728851688829575407993961968, −0.93554787700175185772778571487, 0.5436443816034383996967835394, 1.92333836634974556529788562340, 2.43923191435523843710992678810, 3.6621751295980486788824540198, 4.76529868606314840777845152454, 5.67261988961574228423965040321, 6.903539559311591654109324418687, 7.543745853393237460939405756010, 8.422247919847254156566114581383, 8.899069107939451541008139816201, 10.13589907507390602365760310169, 10.61858826544613554844454283329, 11.35411637914446711027760371501, 12.29369942675706725562918454908, 12.824466563508001230125958176459, 14.13381949725939144276817722439, 14.81128207125013537546688651125, 15.6841508905846260504191149971, 16.54333331263685557032303651228, 17.1805563383162826439771694510, 17.91554286334992514960215538691, 18.61273082261486679198694859611, 19.22415328335123023166477337138, 20.1230591902629837795392894061, 20.915362342212846906173141911319

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