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Properties

Label 1-675-675.103-r1-0-0
Degree $1$
Conductor $675$
Sign $-0.280 - 0.959i$
Analytic cond. $72.5388$
Root an. cond. $72.5388$
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  + (0.694 + 0.719i)2-s + (−0.0348 + 0.999i)4-s + (0.342 + 0.939i)7-s + (−0.743 + 0.669i)8-s + (−0.241 − 0.970i)11-s + (0.694 − 0.719i)13-s + (−0.438 + 0.898i)14-s + (−0.997 − 0.0697i)16-s + (0.207 + 0.978i)17-s + (−0.669 − 0.743i)19-s + (0.529 − 0.848i)22-s + (−0.829 + 0.559i)23-s + 26-s + (−0.951 + 0.309i)28-s + (−0.990 + 0.139i)29-s + ⋯
L(s)  = 1  + (0.694 + 0.719i)2-s + (−0.0348 + 0.999i)4-s + (0.342 + 0.939i)7-s + (−0.743 + 0.669i)8-s + (−0.241 − 0.970i)11-s + (0.694 − 0.719i)13-s + (−0.438 + 0.898i)14-s + (−0.997 − 0.0697i)16-s + (0.207 + 0.978i)17-s + (−0.669 − 0.743i)19-s + (0.529 − 0.848i)22-s + (−0.829 + 0.559i)23-s + 26-s + (−0.951 + 0.309i)28-s + (−0.990 + 0.139i)29-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.280 - 0.959i$
Analytic conductor: \(72.5388\)
Root analytic conductor: \(72.5388\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 675,\ (1:\ ),\ -0.280 - 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2015117317 + 0.2687138765i\)
\(L(\frac12)\) \(\approx\) \(-0.2015117317 + 0.2687138765i\)
\(L(1)\) \(\approx\) \(1.021770440 + 0.6346733982i\)
\(L(1)\) \(\approx\) \(1.021770440 + 0.6346733982i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.694 + 0.719i)T \)
7 \( 1 + (0.342 + 0.939i)T \)
11 \( 1 + (-0.241 - 0.970i)T \)
13 \( 1 + (0.694 - 0.719i)T \)
17 \( 1 + (0.207 + 0.978i)T \)
19 \( 1 + (-0.669 - 0.743i)T \)
23 \( 1 + (-0.829 + 0.559i)T \)
29 \( 1 + (-0.990 + 0.139i)T \)
31 \( 1 + (-0.615 + 0.788i)T \)
37 \( 1 + (0.406 - 0.913i)T \)
41 \( 1 + (-0.719 - 0.694i)T \)
43 \( 1 + (-0.642 + 0.766i)T \)
47 \( 1 + (-0.788 + 0.615i)T \)
53 \( 1 + (-0.951 + 0.309i)T \)
59 \( 1 + (0.241 - 0.970i)T \)
61 \( 1 + (0.961 + 0.275i)T \)
67 \( 1 + (0.139 - 0.990i)T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (0.406 + 0.913i)T \)
79 \( 1 + (-0.990 + 0.139i)T \)
83 \( 1 + (0.469 - 0.882i)T \)
89 \( 1 + (-0.913 + 0.406i)T \)
97 \( 1 + (0.927 + 0.374i)T \)
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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

−22.03120370136753558974727069202, −20.81998259357196732150821127579, −20.616623610982799553376810226317, −19.85348001296863205746906553401, −18.64843565975462875807900925152, −18.2468619587905489616199739948, −16.96647601684814620373681515184, −16.171169487853135193396179813984, −15.023287783518983357690215399720, −14.39017935805442052571692771560, −13.54691383963759831102476139769, −12.88834126428486548632052062473, −11.81117044021580894782328556022, −11.21026965248791072740412883495, −10.16953946094578557161579767898, −9.67128579592747452550304358237, −8.33565295387096888286210408185, −7.18475362130353726638254003660, −6.3456295415399946656441803965, −5.15630410470553624603367469047, −4.29015545338611554423475767328, −3.640079458193000340172627844267, −2.222320677823333352912696924132, −1.430190443879626722544115747269, −0.05586666239421515430782550640, 1.84641219273561864644801591200, 3.073662828301546405949077420377, 3.85271798112145814811540330064, 5.17618668781990649676230315689, 5.77471473950352686803689190799, 6.52509649558518419119809269777, 7.91224125060379172226985941664, 8.38021538336833362105161859185, 9.25220029717937789427317972413, 10.82147453487998023061594569180, 11.4463669196614531537095513184, 12.60730345180475509479203919178, 13.08607045262223084448565807784, 14.12455917086286619167518803088, 14.911899961131337967713121431680, 15.64053763785979622015302966029, 16.27464412706301580529409133172, 17.34302978392073365888282564786, 18.03673650545986100360229832924, 18.87715752269431250387878392012, 19.98390789818885285522540041053, 21.08002153591262717175748862396, 21.644969528374305939787746596260, 22.18606791722297418359333950973, 23.308306547589359556333774436706

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