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Properties

Label 1-675-675.211-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: $\chi_{675} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 675,\ (0:\ ),\ 0.861 + 0.508i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6964943322 + 0.1901038277i\)
\(L(\frac12)\) \(\approx\) \(0.6964943322 + 0.1901038277i\)
\(L(1)\) \(\approx\) \(0.6610855453 + 0.02815135281i\)
\(L(1)\) \(\approx\) \(0.6610855453 + 0.02815135281i\)

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

−22.76432994499745564388730393111, −22.122874258552348634927530932324, −20.546980240834031637606514370639, −20.24416634433226394429337698961, −19.117470829377773497365129812313, −18.81462442638283389731966555503, −17.55670931663942941788731855629, −16.93430573483331341894080653783, −16.33625190289070114102838101625, −15.35876633816462532357009382709, −14.57392822213073082515741045248, −13.79612328564764080045267174527, −12.40188647673173448020823401323, −11.882186995385550977520585966382, −10.432609162065600277143958649436, −9.93865308229952588848195735330, −9.26900960296692937543782315472, −8.07627152174206376714809974461, −7.3288199806294042288727299158, −6.467780857578260402823385721905, −5.676347419510264470497177562457, −4.39969602096960421785325334580, −3.11967542185159759924686191838, −1.928327928948197602815207009054, −0.57864943768036143079830321885, 1.00908114872815304527621913297, 2.30797021453794761939984724460, 3.22791522949782332974303002946, 4.14383128529792224185200945820, 5.740834626551131748794438334480, 6.66939589494043901338196477220, 7.48039340610800594784917200024, 8.585756147025616609280825936, 9.44048351035074595465784540414, 9.87370486823734661125934809123, 11.069349753147749104536599341878, 11.86647736485504381925999473907, 12.54031713205139270188044346096, 13.4818380737712532737131586249, 14.636172559823192261191531191437, 15.64691000753795501788332412851, 16.49891570710942113742322728791, 17.041663905558367089892803908830, 17.95505407558103323344918346620, 19.03415378770409369783460744068, 19.39214125988344368796368885593, 20.044178480278755072459797181045, 21.2572070930767985096822945083, 21.89137236649141158884863652910, 22.44251219486943012375509813072

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