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Properties

Label 1-275-275.233-r0-0-0
Degree $1$
Conductor $275$
Sign $0.962 - 0.269i$
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  + i·2-s + (0.587 − 0.809i)3-s − 4-s + (0.809 + 0.587i)6-s + (−0.951 + 0.309i)7-s − i·8-s + (−0.309 − 0.951i)9-s + (−0.587 + 0.809i)12-s + (0.951 − 0.309i)13-s + (−0.309 − 0.951i)14-s + 16-s + (0.587 − 0.809i)17-s + (0.951 − 0.309i)18-s + 19-s + (−0.309 + 0.951i)21-s + ⋯
L(s)  = 1  + i·2-s + (0.587 − 0.809i)3-s − 4-s + (0.809 + 0.587i)6-s + (−0.951 + 0.309i)7-s − i·8-s + (−0.309 − 0.951i)9-s + (−0.587 + 0.809i)12-s + (0.951 − 0.309i)13-s + (−0.309 − 0.951i)14-s + 16-s + (0.587 − 0.809i)17-s + (0.951 − 0.309i)18-s + 19-s + (−0.309 + 0.951i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.962 - 0.269i$
Analytic conductor: \(1.27709\)
Root analytic conductor: \(1.27709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (0:\ ),\ 0.962 - 0.269i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.180009941 - 0.1622674704i\)
\(L(\frac12)\) \(\approx\) \(1.180009941 - 0.1622674704i\)
\(L(1)\) \(\approx\) \(1.072958914 + 0.08601377258i\)
\(L(1)\) \(\approx\) \(1.072958914 + 0.08601377258i\)

Euler product

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

−26.17844728839274538117253712867, −25.11717125626021572569548580679, −23.49011054677734114568315461340, −22.81991969157657058240347713209, −21.78436146020762299410176779736, −21.220825615804722462409404977607, −20.18756432259430911369965003915, −19.588573002916998504325552166988, −18.780121706212432831122961785620, −17.56222083065976757132086351040, −16.38043004500342462089414384334, −15.61978859269841846551526959270, −14.20928027039018373891521677939, −13.66720273267616922422428097992, −12.62108072650533464619430382122, −11.44439947258627727687335777246, −10.38788376511917643142204964874, −9.77761368406635564842739435865, −8.86054554576920309475037905242, −7.85926706370479065473826608309, −6.076948763590623120677832014, −4.77673383303869127150625370389, −3.58913072064193235750077776965, −3.134639090449845914273026743714, −1.50926765109618624120808279303, 0.85114675394197331098200136488, 2.813740965799376895995920824637, 3.836442790535793650288895112628, 5.5456313923113294723490088300, 6.38357314148270157327078169380, 7.287182940187945245389332066335, 8.27122066570194215166859225002, 9.14071222197623827846495765445, 10.0588453134388787690499805669, 11.94675729821399172761909322999, 12.81848852459560085638379253573, 13.69985688860557621871536492156, 14.33826213116641449699539197276, 15.64203234645134026859598208902, 16.13405620426928661691695751907, 17.45833067339780340860771738254, 18.40710942237039582697833163499, 18.879314556671527698717003474228, 19.99422505462120980617125766104, 21.06611366723033429609202899570, 22.58807251859061368943893664320, 22.914586375772479702783898538109, 24.11584117197388124503681257641, 24.77733632007090014874657295565, 25.67864489047306299937972570212

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