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Properties

Label 1-275-275.92-r1-0-0
Degree $1$
Conductor $275$
Sign $-0.390 + 0.920i$
Analytic cond. $29.5528$
Root an. cond. $29.5528$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.390 + 0.920i$
Analytic conductor: \(29.5528\)
Root analytic conductor: \(29.5528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (1:\ ),\ -0.390 + 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3443257956 + 0.5198463311i\)
\(L(\frac12)\) \(\approx\) \(0.3443257956 + 0.5198463311i\)
\(L(1)\) \(\approx\) \(1.119592058 - 0.2060814316i\)
\(L(1)\) \(\approx\) \(1.119592058 - 0.2060814316i\)

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

−24.90671403379996247827323561749, −24.52353619569602538713959487186, −23.5452654484594981705515967766, −22.56252674196263611184540272617, −21.7921473742252067640109499884, −20.69811104886674528672581260737, −19.61231119041282494835726543105, −18.63920629656503132094113715384, −17.86017190261279283040391439443, −16.76304081263684610216862209231, −15.73452620762686617119599070542, −14.798038215987095660829096709792, −14.132641664488092415419610411478, −12.93382303598869255688677376154, −12.51843468263845879228549937151, −11.439865263043116778307874169742, −9.31881104504977941128991886950, −8.893560071889453786854204827785, −7.51206968999954011183468392084, −6.88238725965273127831285186918, −5.83821309004623937687953639248, −4.65022945178356647933202226029, −3.114278103588491348132211433187, −2.37718937436604249424597454603, −0.13242525399660787185184044486, 1.73053524486948349061987011161, 3.09551897496254128979124714553, 3.84106325335131860788207931390, 4.82820230052609192968724096841, 6.01090672187221559881009335340, 7.53429709517917880436991295494, 8.881877330596136560389471804956, 10.05397180957904530687149895594, 10.30523021327195257703245963789, 11.5512034244637228985014683566, 12.827107049228489105770919544794, 13.526267011202103858253075059653, 14.61444807702110867024866932713, 15.1646008881959441587152023365, 16.418127860643740186656066460628, 17.363193103396694514557484550032, 18.992135997082081767123153484079, 19.59403160593145160188475099730, 20.31264973497517436808322100165, 21.19649127454283024504983107765, 21.98375400976357132579909232403, 22.79486322643244404050846822020, 23.64691899634171792297210201026, 24.83604435953222288505031620208, 25.83742367680770734409537245430

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