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Properties

Label 1-3525-3525.128-r0-0-0
Degree $1$
Conductor $3525$
Sign $-0.528 + 0.848i$
Analytic cond. $16.3700$
Root an. cond. $16.3700$
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  + (0.565 + 0.824i)2-s + (−0.360 + 0.932i)4-s + (−0.816 + 0.576i)7-s + (−0.973 + 0.230i)8-s + (−0.149 + 0.988i)11-s + (0.190 − 0.981i)13-s + (−0.937 − 0.347i)14-s + (−0.740 − 0.672i)16-s + (0.711 − 0.702i)17-s + (−0.927 − 0.373i)19-s + (−0.900 + 0.435i)22-s + (0.959 + 0.282i)23-s + (0.917 − 0.398i)26-s + (−0.243 − 0.969i)28-s + (0.905 + 0.423i)29-s + ⋯
L(s)  = 1  + (0.565 + 0.824i)2-s + (−0.360 + 0.932i)4-s + (−0.816 + 0.576i)7-s + (−0.973 + 0.230i)8-s + (−0.149 + 0.988i)11-s + (0.190 − 0.981i)13-s + (−0.937 − 0.347i)14-s + (−0.740 − 0.672i)16-s + (0.711 − 0.702i)17-s + (−0.927 − 0.373i)19-s + (−0.900 + 0.435i)22-s + (0.959 + 0.282i)23-s + (0.917 − 0.398i)26-s + (−0.243 − 0.969i)28-s + (0.905 + 0.423i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.528 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.528 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.528 + 0.848i$
Analytic conductor: \(16.3700\)
Root analytic conductor: \(16.3700\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3525,\ (0:\ ),\ -0.528 + 0.848i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9187689630 + 1.654485900i\)
\(L(\frac12)\) \(\approx\) \(0.9187689630 + 1.654485900i\)
\(L(1)\) \(\approx\) \(1.023362684 + 0.7254908906i\)
\(L(1)\) \(\approx\) \(1.023362684 + 0.7254908906i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
47 \( 1 \)
good2 \( 1 + (0.565 + 0.824i)T \)
7 \( 1 + (-0.816 + 0.576i)T \)
11 \( 1 + (-0.149 + 0.988i)T \)
13 \( 1 + (0.190 - 0.981i)T \)
17 \( 1 + (0.711 - 0.702i)T \)
19 \( 1 + (-0.927 - 0.373i)T \)
23 \( 1 + (0.959 + 0.282i)T \)
29 \( 1 + (0.905 + 0.423i)T \)
31 \( 1 + (0.994 + 0.109i)T \)
37 \( 1 + (0.347 + 0.937i)T \)
41 \( 1 + (0.996 - 0.0818i)T \)
43 \( 1 + (0.631 - 0.775i)T \)
53 \( 1 + (0.973 + 0.230i)T \)
59 \( 1 + (0.256 - 0.966i)T \)
61 \( 1 + (-0.937 - 0.347i)T \)
67 \( 1 + (0.800 - 0.598i)T \)
71 \( 1 + (-0.792 + 0.609i)T \)
73 \( 1 + (-0.216 + 0.976i)T \)
79 \( 1 + (0.176 + 0.984i)T \)
83 \( 1 + (-0.163 - 0.986i)T \)
89 \( 1 + (-0.531 + 0.847i)T \)
97 \( 1 + (-0.109 - 0.994i)T \)
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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

−18.911003976725343147039036371144, −18.001651759941907476168363231554, −17.01193804321170416343462172477, −16.445788599778129280253840536340, −15.75750036527462041988607831096, −14.781374251559042359841513879121, −14.23761892946017947050678831891, −13.51553951622222711024191915011, −13.012971394329198847777825712713, −12.29423121325479501396413714677, −11.58691161809813143769040493182, −10.679001855925534828731180264135, −10.44147812631866390699508676454, −9.46684490605048368231610300911, −8.867382522254609702535043612516, −8.00300500904910342534450356232, −6.839088579427595924867148988214, −6.17850515120642249221742742765, −5.69272958105771369972547432373, −4.38641075550241615563418276838, −4.08974778961350491691425109802, −3.13071937261798682164217252125, −2.552265126509397604017425728246, −1.38396694493789566471366159241, −0.614472226856880231600993813784, 0.84528094378542188582066724797, 2.5210326218551182797575672589, 2.8915079778059503722517230661, 3.82901112290799856438212496144, 4.79239967965170498338128612179, 5.309477278950491295370574261920, 6.162099802549193017079458371659, 6.80002541600786771919224282299, 7.477689106204116819964988442942, 8.28791409012641594125713275116, 8.995942700195205445607744170359, 9.73803651935894474015288448779, 10.46339361233492454877460358800, 11.583348453670601599369308298824, 12.36606326722121908395865102537, 12.76088377674250550072522023215, 13.38032480178713241841288873763, 14.237030887356301516339115411135, 15.008271288031320478078190505475, 15.55916630388607282828508085814, 15.90924863865041362789021742669, 16.940418422134335948194738461141, 17.39165657858562766362655959725, 18.174665961985693502926876824864, 18.79334802025469653776228750793

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