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Properties

Label 1-675-675.139-r0-0-0
Degree $1$
Conductor $675$
Sign $0.00930 - 0.999i$
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  + (−0.848 + 0.529i)2-s + (0.438 − 0.898i)4-s + (−0.173 + 0.984i)7-s + (0.104 + 0.994i)8-s + (0.0348 + 0.999i)11-s + (−0.848 − 0.529i)13-s + (−0.374 − 0.927i)14-s + (−0.615 − 0.788i)16-s + (−0.913 − 0.406i)17-s + (−0.104 − 0.994i)19-s + (−0.559 − 0.829i)22-s + (−0.990 − 0.139i)23-s + 26-s + (0.809 + 0.587i)28-s + (−0.241 − 0.970i)29-s + ⋯
L(s)  = 1  + (−0.848 + 0.529i)2-s + (0.438 − 0.898i)4-s + (−0.173 + 0.984i)7-s + (0.104 + 0.994i)8-s + (0.0348 + 0.999i)11-s + (−0.848 − 0.529i)13-s + (−0.374 − 0.927i)14-s + (−0.615 − 0.788i)16-s + (−0.913 − 0.406i)17-s + (−0.104 − 0.994i)19-s + (−0.559 − 0.829i)22-s + (−0.990 − 0.139i)23-s + 26-s + (0.809 + 0.587i)28-s + (−0.241 − 0.970i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00930 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00930 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.00930 - 0.999i$
Analytic conductor: \(3.13468\)
Root analytic conductor: \(3.13468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 675,\ (0:\ ),\ 0.00930 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2012827203 - 0.1994177621i\)
\(L(\frac12)\) \(\approx\) \(0.2012827203 - 0.1994177621i\)
\(L(1)\) \(\approx\) \(0.5386109102 + 0.1028584145i\)
\(L(1)\) \(\approx\) \(0.5386109102 + 0.1028584145i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.848 + 0.529i)T \)
7 \( 1 + (-0.173 + 0.984i)T \)
11 \( 1 + (0.0348 + 0.999i)T \)
13 \( 1 + (-0.848 - 0.529i)T \)
17 \( 1 + (-0.913 - 0.406i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (-0.990 - 0.139i)T \)
29 \( 1 + (-0.241 - 0.970i)T \)
31 \( 1 + (-0.719 - 0.694i)T \)
37 \( 1 + (-0.669 + 0.743i)T \)
41 \( 1 + (0.848 + 0.529i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (0.719 - 0.694i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (0.0348 - 0.999i)T \)
61 \( 1 + (-0.882 - 0.469i)T \)
67 \( 1 + (0.241 - 0.970i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (-0.669 - 0.743i)T \)
79 \( 1 + (-0.241 - 0.970i)T \)
83 \( 1 + (0.997 + 0.0697i)T \)
89 \( 1 + (0.669 + 0.743i)T \)
97 \( 1 + (-0.961 - 0.275i)T \)
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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

−22.759652246193054108200355833045, −21.92307193171473408606510958885, −21.249499971563297800247346311365, −20.30019249451425104109596555712, −19.60357422343143794412998072117, −19.080259820109606613932026912096, −18.00211668828549531815745463907, −17.31336186843605256902084821802, −16.4121915609024750023981196715, −16.05819347956319751553487876925, −14.54457061498612405079931082650, −13.76214058354000158154230726116, −12.755775034915239387687767522371, −11.98917713189856537895219054118, −10.869323686444185828085166381472, −10.50027406013580553985950696104, −9.413442869167588406822034333128, −8.64994307223101551731605732889, −7.627812731896061139335130973642, −6.94325493376608177605107255637, −5.84991902194875672384012735650, −4.230745983328097486261142542229, −3.59400125048563062680834886119, −2.34465881061300123609852210159, −1.257402745718686308965750592187, 0.18395037104452178084614403491, 2.0430389151494575289811355424, 2.57417524859053701606711584368, 4.463317050395295388143542793426, 5.35843189514603451490721400338, 6.29806904765350963081728291687, 7.21218676569625999724739351139, 8.003195431952804214979158988335, 9.11603886380615157296356835768, 9.56606568185475100308551296133, 10.54348111010646729154916444486, 11.595646094878680463674909418666, 12.35890035976335273263862215811, 13.46856522377336172261638968942, 14.655028782014617583876988652997, 15.35311891164860152765795017675, 15.75956104241353235077744911921, 16.97862686394468658813192920164, 17.69218564173921487896751706152, 18.258614018366458972414229644678, 19.20336128543622657009970160275, 19.96511035921903674962184756667, 20.59176732922214304077380915000, 21.95254825489419645859743753974, 22.49377096746491883919048147190

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