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Properties

Label 1-675-675.202-r1-0-0
Degree $1$
Conductor $675$
Sign $0.508 - 0.861i$
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  + (−0.999 + 0.0348i)2-s + (0.997 − 0.0697i)4-s + (−0.642 + 0.766i)7-s + (−0.994 + 0.104i)8-s + (−0.882 − 0.469i)11-s + (−0.999 − 0.0348i)13-s + (0.615 − 0.788i)14-s + (0.990 − 0.139i)16-s + (−0.406 + 0.913i)17-s + (0.104 + 0.994i)19-s + (0.898 + 0.438i)22-s + (−0.927 + 0.374i)23-s + 26-s + (−0.587 + 0.809i)28-s + (−0.961 − 0.275i)29-s + ⋯
L(s)  = 1  + (−0.999 + 0.0348i)2-s + (0.997 − 0.0697i)4-s + (−0.642 + 0.766i)7-s + (−0.994 + 0.104i)8-s + (−0.882 − 0.469i)11-s + (−0.999 − 0.0348i)13-s + (0.615 − 0.788i)14-s + (0.990 − 0.139i)16-s + (−0.406 + 0.913i)17-s + (0.104 + 0.994i)19-s + (0.898 + 0.438i)22-s + (−0.927 + 0.374i)23-s + 26-s + (−0.587 + 0.809i)28-s + (−0.961 − 0.275i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.508 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.508 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.508 - 0.861i$
Analytic conductor: \(72.5388\)
Root analytic conductor: \(72.5388\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 675,\ (1:\ ),\ 0.508 - 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2204694134 - 0.1259235835i\)
\(L(\frac12)\) \(\approx\) \(0.2204694134 - 0.1259235835i\)
\(L(1)\) \(\approx\) \(0.4762378372 + 0.08222082533i\)
\(L(1)\) \(\approx\) \(0.4762378372 + 0.08222082533i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.999 + 0.0348i)T \)
7 \( 1 + (-0.642 + 0.766i)T \)
11 \( 1 + (-0.882 - 0.469i)T \)
13 \( 1 + (-0.999 - 0.0348i)T \)
17 \( 1 + (-0.406 + 0.913i)T \)
19 \( 1 + (0.104 + 0.994i)T \)
23 \( 1 + (-0.927 + 0.374i)T \)
29 \( 1 + (-0.961 - 0.275i)T \)
31 \( 1 + (-0.241 + 0.970i)T \)
37 \( 1 + (0.743 + 0.669i)T \)
41 \( 1 + (0.0348 - 0.999i)T \)
43 \( 1 + (-0.984 - 0.173i)T \)
47 \( 1 + (0.970 - 0.241i)T \)
53 \( 1 + (-0.587 + 0.809i)T \)
59 \( 1 + (0.882 - 0.469i)T \)
61 \( 1 + (0.848 - 0.529i)T \)
67 \( 1 + (0.275 + 0.961i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (0.743 - 0.669i)T \)
79 \( 1 + (-0.961 - 0.275i)T \)
83 \( 1 + (-0.829 - 0.559i)T \)
89 \( 1 + (-0.669 - 0.743i)T \)
97 \( 1 + (-0.694 - 0.719i)T \)
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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

−22.61596606762145820663876330349, −21.83642514093998046637373745241, −20.66900064378664549202173263150, −20.09695171971414357852879751882, −19.52541296415753650050534396796, −18.44850220545118273270925250973, −17.87257403549381052589466456404, −16.93643867445523551434594634686, −16.27997829651563120512520710498, −15.4978897407879155267483857601, −14.6079839648546053572017189506, −13.36208819998687179422894276371, −12.63222612443827612045526927872, −11.54479311277611386834875969036, −10.76154923458398213928195467706, −9.75328576015039941181811120274, −9.46742438783175676626967536075, −8.07909515618268004584942909554, −7.315768365597473739890319445091, −6.73474127581168490135580714566, −5.46058382624520036186764096429, −4.26458842023515881649710661333, −2.887863963510839164926387158490, −2.16519490326696177131217498413, −0.56521197892007299667210909282, 0.13834501773241268456619095244, 1.78648922491397002580916928649, 2.62492315826570656586625663269, 3.67526913212474521423946321222, 5.41723769372390521804551007677, 6.0260366692172115322953043640, 7.12014588278640044670039342315, 8.05720995467585696880268448940, 8.74754557429950864402504307707, 9.820406409537200003818847659241, 10.29598259051537678568052589083, 11.40876160401481186038041259920, 12.28473119982515131471820363441, 12.9665960530358454166234958435, 14.34870726915459880648205649028, 15.28660247074540059486418890892, 15.88682977435678601747793253477, 16.71790423952454614441047046947, 17.5155245617922175728022512246, 18.517743878703844920534446994155, 18.94031028832270438609988201036, 19.80376113040399647744349270757, 20.57771009097777461797058616518, 21.65573324021809295721820404366, 22.09014098539430436709532566031

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