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Properties

Label 1-675-675.259-r0-0-0
Degree $1$
Conductor $675$
Sign $0.861 - 0.508i$
Analytic cond. $3.13468$
Root an. cond. $3.13468$
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  + (0.882 − 0.469i)2-s + (0.559 − 0.829i)4-s + (0.939 + 0.342i)7-s + (0.104 − 0.994i)8-s + (0.848 + 0.529i)11-s + (0.882 + 0.469i)13-s + (0.990 − 0.139i)14-s + (−0.374 − 0.927i)16-s + (−0.913 + 0.406i)17-s + (−0.104 + 0.994i)19-s + (0.997 + 0.0697i)22-s + (0.615 + 0.788i)23-s + 26-s + (0.809 − 0.587i)28-s + (−0.719 − 0.694i)29-s + ⋯
L(s)  = 1  + (0.882 − 0.469i)2-s + (0.559 − 0.829i)4-s + (0.939 + 0.342i)7-s + (0.104 − 0.994i)8-s + (0.848 + 0.529i)11-s + (0.882 + 0.469i)13-s + (0.990 − 0.139i)14-s + (−0.374 − 0.927i)16-s + (−0.913 + 0.406i)17-s + (−0.104 + 0.994i)19-s + (0.997 + 0.0697i)22-s + (0.615 + 0.788i)23-s + 26-s + (0.809 − 0.587i)28-s + (−0.719 − 0.694i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.861 - 0.508i$
Analytic conductor: \(3.13468\)
Root analytic conductor: \(3.13468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 675,\ (0:\ ),\ 0.861 - 0.508i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.762258445 - 0.7539413882i\)
\(L(\frac12)\) \(\approx\) \(2.762258445 - 0.7539413882i\)
\(L(1)\) \(\approx\) \(1.942709355 - 0.4533009111i\)
\(L(1)\) \(\approx\) \(1.942709355 - 0.4533009111i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.882 - 0.469i)T \)
7 \( 1 + (0.939 + 0.342i)T \)
11 \( 1 + (0.848 + 0.529i)T \)
13 \( 1 + (0.882 + 0.469i)T \)
17 \( 1 + (-0.913 + 0.406i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (0.615 + 0.788i)T \)
29 \( 1 + (-0.719 - 0.694i)T \)
31 \( 1 + (0.961 + 0.275i)T \)
37 \( 1 + (-0.669 - 0.743i)T \)
41 \( 1 + (-0.882 - 0.469i)T \)
43 \( 1 + (-0.766 + 0.642i)T \)
47 \( 1 + (-0.961 + 0.275i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.848 - 0.529i)T \)
61 \( 1 + (0.0348 - 0.999i)T \)
67 \( 1 + (0.719 - 0.694i)T \)
71 \( 1 + (-0.104 - 0.994i)T \)
73 \( 1 + (-0.669 + 0.743i)T \)
79 \( 1 + (-0.719 - 0.694i)T \)
83 \( 1 + (-0.438 + 0.898i)T \)
89 \( 1 + (0.669 - 0.743i)T \)
97 \( 1 + (0.241 - 0.970i)T \)
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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

−22.80469633562564193899463501269, −22.116705829125009318509226488818, −21.32150537945372011293579950078, −20.47427136514726196188267870158, −19.95523381403429783737790551730, −18.60494891010263299682113767295, −17.6343992864506466150010786875, −17.01490682651188912771405926260, −16.12988683539932587235519979825, −15.19252780521166542738747674956, −14.61145954788056638062657010327, −13.52635275618872119366597714067, −13.274539332206901243937838529881, −11.82343940920026464981123010656, −11.339577762764698579831463721900, −10.51457722270727355145555055413, −8.766298779793409026175898213449, −8.41526969025726755206392904522, −7.09462775574903430635124196305, −6.52995584968496063550570617047, −5.35747688755287101929877869563, −4.57324532498435459828866831554, −3.6840562001207333813264387787, −2.60346664584312360357850874862, −1.26902055401810138508715588531, 1.497718707919327521993289265193, 1.9823965426151065199193493717, 3.50519595143591922825080533974, 4.24845707925616089214483222506, 5.161338887566840101011281270399, 6.15700599220203293593509832501, 6.95340713550200998741505802006, 8.23831652548530561213031179680, 9.20859891450050672789265900502, 10.2302237482380787990463407271, 11.31518201381730902874236072429, 11.64384176398676948768282974419, 12.65910714815773428514825432721, 13.567619458611965780734630704336, 14.34584841600938150692290855276, 15.0601123135294273690186718340, 15.76016827006552620675420610314, 16.936101062736827055407779288779, 17.83315735890884807354259530638, 18.80331012859509516414268325707, 19.53741446081031159279068395628, 20.50072083884887232137890650534, 21.10271621385843654685324016093, 21.75348931437269995024022046202, 22.74555365519081184681203926460

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