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Properties

Label 1-675-675.248-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  + (0.469 − 0.882i)2-s + (−0.559 − 0.829i)4-s + (0.342 + 0.939i)7-s + (−0.994 + 0.104i)8-s + (−0.848 + 0.529i)11-s + (−0.469 − 0.882i)13-s + (0.990 + 0.139i)14-s + (−0.374 + 0.927i)16-s + (−0.406 + 0.913i)17-s + (0.104 + 0.994i)19-s + (0.0697 + 0.997i)22-s + (0.788 + 0.615i)23-s − 26-s + (0.587 − 0.809i)28-s + (−0.719 + 0.694i)29-s + ⋯
L(s)  = 1  + (0.469 − 0.882i)2-s + (−0.559 − 0.829i)4-s + (0.342 + 0.939i)7-s + (−0.994 + 0.104i)8-s + (−0.848 + 0.529i)11-s + (−0.469 − 0.882i)13-s + (0.990 + 0.139i)14-s + (−0.374 + 0.927i)16-s + (−0.406 + 0.913i)17-s + (0.104 + 0.994i)19-s + (0.0697 + 0.997i)22-s + (0.788 + 0.615i)23-s − 26-s + (0.587 − 0.809i)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: $\chi_{675} (248, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 675,\ (0:\ ),\ 0.861 + 0.508i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.109571368 + 0.3028506545i\)
\(L(\frac12)\) \(\approx\) \(1.109571368 + 0.3028506545i\)
\(L(1)\) \(\approx\) \(1.052452552 - 0.2357378730i\)
\(L(1)\) \(\approx\) \(1.052452552 - 0.2357378730i\)

Euler product

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

−22.85301005627265893452704643789, −22.02010687591772009209541216661, −21.10001826121385376124112099949, −20.56053980651669232187764078558, −19.3083130818638550497993155413, −18.41373436962564616084036053757, −17.51183227118181042022966021498, −16.84346186520778311219843882557, −16.08460128054751703369612032662, −15.28253262751429961749208348757, −14.32405668209133684911382164889, −13.59260807608423438035635858821, −13.11449447539092735852792952305, −11.816169981683450775660971250918, −11.08206088182942666362992258487, −9.8743358511401019763255519435, −8.88382823975861050455168118783, −7.98392073963385671229745352236, −7.09551358346482251682093613605, −6.531407912547102510417976014159, −5.0561602732669709934424669759, −4.69829377243321753889044577906, −3.49540285332554854685672874559, −2.418358039336720656908321903921, −0.47514585195213403618428269152, 1.467384138163045344635701252040, 2.41621001410184328318243138649, 3.26629055874300227331730492076, 4.50951663674438629685107769356, 5.37853068663929045561508801523, 5.99921376899002978771521200787, 7.553282446411579674343771952707, 8.50451141820962504404994572459, 9.460717365652609232123231874538, 10.35938593557390353261081654174, 11.04657969735567468130634394914, 12.14207334320507642958954473558, 12.662806967873730232032951667, 13.40287292121603678907045098150, 14.719112532214968577553983021158, 15.0318271291486059168245110927, 15.955092693884781198718831756719, 17.506510765591876022017323873153, 17.962975651066273349852722750067, 18.94342183714323815808974822358, 19.52984055220066572258202708160, 20.67979440767293817398752708712, 21.03634585766870957394630044785, 21.99000002887571046912430662269, 22.65165984164256319308031475559

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