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Properties

Label 1-675-675.121-r0-0-0
Degree $1$
Conductor $675$
Sign $0.280 + 0.959i$
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  + (0.961 + 0.275i)2-s + (0.848 + 0.529i)4-s + (0.766 + 0.642i)7-s + (0.669 + 0.743i)8-s + (−0.719 + 0.694i)11-s + (0.961 − 0.275i)13-s + (0.559 + 0.829i)14-s + (0.438 + 0.898i)16-s + (−0.978 + 0.207i)17-s + (0.669 + 0.743i)19-s + (−0.882 + 0.469i)22-s + (−0.997 + 0.0697i)23-s + 26-s + (0.309 + 0.951i)28-s + (−0.615 − 0.788i)29-s + ⋯
L(s)  = 1  + (0.961 + 0.275i)2-s + (0.848 + 0.529i)4-s + (0.766 + 0.642i)7-s + (0.669 + 0.743i)8-s + (−0.719 + 0.694i)11-s + (0.961 − 0.275i)13-s + (0.559 + 0.829i)14-s + (0.438 + 0.898i)16-s + (−0.978 + 0.207i)17-s + (0.669 + 0.743i)19-s + (−0.882 + 0.469i)22-s + (−0.997 + 0.0697i)23-s + 26-s + (0.309 + 0.951i)28-s + (−0.615 − 0.788i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.280 + 0.959i$
Analytic conductor: \(3.13468\)
Root analytic conductor: \(3.13468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 675,\ (0:\ ),\ 0.280 + 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.234763236 + 1.675875527i\)
\(L(\frac12)\) \(\approx\) \(2.234763236 + 1.675875527i\)
\(L(1)\) \(\approx\) \(1.847852649 + 0.7075341596i\)
\(L(1)\) \(\approx\) \(1.847852649 + 0.7075341596i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.961 + 0.275i)T \)
7 \( 1 + (0.766 + 0.642i)T \)
11 \( 1 + (-0.719 + 0.694i)T \)
13 \( 1 + (0.961 - 0.275i)T \)
17 \( 1 + (-0.978 + 0.207i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (-0.997 + 0.0697i)T \)
29 \( 1 + (-0.615 - 0.788i)T \)
31 \( 1 + (-0.374 - 0.927i)T \)
37 \( 1 + (0.913 + 0.406i)T \)
41 \( 1 + (0.961 - 0.275i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (-0.374 + 0.927i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.719 - 0.694i)T \)
61 \( 1 + (-0.241 - 0.970i)T \)
67 \( 1 + (-0.615 + 0.788i)T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (0.913 - 0.406i)T \)
79 \( 1 + (-0.615 - 0.788i)T \)
83 \( 1 + (0.0348 + 0.999i)T \)
89 \( 1 + (0.913 - 0.406i)T \)
97 \( 1 + (0.990 - 0.139i)T \)
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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

−22.638738289152130142022525847924, −21.56695794696592003821437115357, −21.2076441693606763276830540663, −20.1211091679112597750073051726, −19.81156980769217328867240491995, −18.38860275799782132164361071319, −17.903514622978877796581400239754, −16.39831101930429388589762987803, −16.02577030588081277689751043406, −14.980032153676571791482215488832, −14.05102292014501840854833341361, −13.529460133214475754875845548798, −12.790496367630871045382494644492, −11.42632569204270177346144854062, −11.14423429651270689662421206068, −10.301414556913533146563589838, −8.99753604484264412411323140718, −7.89323243194042543856044390256, −6.99806857663818545406446499819, −5.994191684603517551822806979131, −5.07182655972341478455498009199, −4.222933414078965346900455921625, −3.29544785818881314168418326708, −2.16280052106077523056792856144, −1.03246585107359376936779786252, 1.75612751826235531722823529110, 2.50770265345291071096918005283, 3.816529248620739671302206531969, 4.628583591293589108525315517184, 5.65548368650554640336197737341, 6.21409821689413210341270800603, 7.672977383285441471460135541643, 8.01915916560557892556178075672, 9.29462582763006908156194368035, 10.6072485695637748627913118229, 11.34044881041348780291052752957, 12.164804633466639036650668645, 13.00681160011863905770616271490, 13.77203462714666907659329286839, 14.68476059259563852081969219312, 15.48400598126257513800882787852, 15.91483328303738499615609066661, 17.122280917262194969253537191977, 17.97804969460503034744746695844, 18.65696285257029491265301013712, 20.139787447420013645911379533522, 20.58552258863855406044262494087, 21.32740676480563122830252857903, 22.22873500811485532118619316690, 22.84422082880452461944323743508

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