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Properties

Label 1-275-275.272-r0-0-0
Degree $1$
Conductor $275$
Sign $-0.582 - 0.812i$
Analytic cond. $1.27709$
Root an. cond. $1.27709$
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  + (0.587 + 0.809i)2-s + (−0.587 + 0.809i)3-s + (−0.309 + 0.951i)4-s − 6-s + (−0.587 + 0.809i)7-s + (−0.951 + 0.309i)8-s + (−0.309 − 0.951i)9-s + (−0.587 − 0.809i)12-s + (−0.951 + 0.309i)13-s − 14-s + (−0.809 − 0.587i)16-s − i·17-s + (0.587 − 0.809i)18-s + (0.309 + 0.951i)19-s + (−0.309 − 0.951i)21-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (−0.587 + 0.809i)3-s + (−0.309 + 0.951i)4-s − 6-s + (−0.587 + 0.809i)7-s + (−0.951 + 0.309i)8-s + (−0.309 − 0.951i)9-s + (−0.587 − 0.809i)12-s + (−0.951 + 0.309i)13-s − 14-s + (−0.809 − 0.587i)16-s − i·17-s + (0.587 − 0.809i)18-s + (0.309 + 0.951i)19-s + (−0.309 − 0.951i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.582 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.582 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.582 - 0.812i$
Analytic conductor: \(1.27709\)
Root analytic conductor: \(1.27709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (0:\ ),\ -0.582 - 0.812i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3016841079 + 0.5872157173i\)
\(L(\frac12)\) \(\approx\) \(-0.3016841079 + 0.5872157173i\)
\(L(1)\) \(\approx\) \(0.4783399086 + 0.6878407940i\)
\(L(1)\) \(\approx\) \(0.4783399086 + 0.6878407940i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.587 + 0.809i)T \)
3 \( 1 + (-0.587 + 0.809i)T \)
7 \( 1 + (-0.587 + 0.809i)T \)
13 \( 1 + (-0.951 + 0.309i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.587 + 0.809i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 - iT \)
59 \( 1 - T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (0.951 + 0.309i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.951 + 0.309i)T \)
79 \( 1 + T \)
83 \( 1 + iT \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + iT \)
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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

−24.70876636904180672301009283580, −23.86961419123225299124956527599, −23.3316535063259743109904130028, −22.23717037977342075062357529844, −21.87176975742600162797250297229, −20.24545539456696095401058593486, −19.73254489144745871527864166692, −18.91811523836733381702321651280, −17.84145461676610802017236508146, −17.02155368369129342421921023343, −15.820272926644337289598701986548, −14.44959800358667193412972341222, −13.66962019264326516782018739159, −12.636000173466987045419562621945, −12.28228568863258887616295889316, −10.83850626559275377001564470472, −10.41861777834278027768223438510, −9.03577459062944039680172375851, −7.45199953528982465850419016594, −6.54073257641131346368236995232, −5.468848025665932532117452681691, −4.37928996757082978300101224219, −3.03794902881634107418941691263, −1.80413255918967329094259847983, −0.3793442556124203010967167318, 2.710497951804625146785680086822, 3.831773098327347109553207033686, 4.97505419106794403530436704268, 5.747266639231345086830622039229, 6.67646944495354385692074716966, 7.94167401730182578035003776814, 9.30487502480799112795296342213, 9.85791217789192456845645005090, 11.699139221868059307298987943717, 12.03004097134321304645859214098, 13.2978239604912892206425912851, 14.505887245987907737734993866804, 15.254669236252010100354831176847, 16.12301827916034755892762244913, 16.70557264047213995225447279079, 17.75627480271908029722475552369, 18.676228957809700682599409558407, 20.18123843081228734555731319572, 21.25905431191082209654868330116, 21.97564472011965905496361779755, 22.59017116654660949182258406383, 23.38361036133489344165169842322, 24.487801619529625513856840738216, 25.279651520430143517775517438392, 26.2379886571380576869301562886

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