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Properties

Label 1-1575-1575.1042-r0-0-0
Degree $1$
Conductor $1575$
Sign $-0.201 + 0.979i$
Analytic cond. $7.31426$
Root an. cond. $7.31426$
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  + (−0.406 + 0.913i)2-s + (−0.669 − 0.743i)4-s + (0.951 − 0.309i)8-s + (0.913 + 0.406i)11-s + (0.406 + 0.913i)13-s + (−0.104 + 0.994i)16-s + (0.951 − 0.309i)17-s + (0.309 + 0.951i)19-s + (−0.743 + 0.669i)22-s + (−0.994 + 0.104i)23-s − 26-s + (0.978 − 0.207i)29-s + (0.978 + 0.207i)31-s + (−0.866 − 0.5i)32-s + (−0.104 + 0.994i)34-s + ⋯
L(s)  = 1  + (−0.406 + 0.913i)2-s + (−0.669 − 0.743i)4-s + (0.951 − 0.309i)8-s + (0.913 + 0.406i)11-s + (0.406 + 0.913i)13-s + (−0.104 + 0.994i)16-s + (0.951 − 0.309i)17-s + (0.309 + 0.951i)19-s + (−0.743 + 0.669i)22-s + (−0.994 + 0.104i)23-s − 26-s + (0.978 − 0.207i)29-s + (0.978 + 0.207i)31-s + (−0.866 − 0.5i)32-s + (−0.104 + 0.994i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.201 + 0.979i$
Analytic conductor: \(7.31426\)
Root analytic conductor: \(7.31426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1042, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1575,\ (0:\ ),\ -0.201 + 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8274344469 + 1.014534866i\)
\(L(\frac12)\) \(\approx\) \(0.8274344469 + 1.014534866i\)
\(L(1)\) \(\approx\) \(0.8204361839 + 0.4539915773i\)
\(L(1)\) \(\approx\) \(0.8204361839 + 0.4539915773i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.406 + 0.913i)T \)
11 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (0.406 + 0.913i)T \)
17 \( 1 + (0.951 - 0.309i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.994 + 0.104i)T \)
29 \( 1 + (0.978 - 0.207i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (0.587 - 0.809i)T \)
41 \( 1 + (-0.913 + 0.406i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (0.207 + 0.978i)T \)
53 \( 1 + (-0.951 - 0.309i)T \)
59 \( 1 + (0.913 - 0.406i)T \)
61 \( 1 + (-0.913 - 0.406i)T \)
67 \( 1 + (0.207 - 0.978i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.587 - 0.809i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (0.743 + 0.669i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.207 + 0.978i)T \)
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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

−20.1896356405007006606381009487, −19.65670126238979620174705739457, −18.881592085731401090597860706632, −18.17419074464602037223049181458, −17.429493208823325061055484136924, −16.82383091008355481251159835443, −15.955451382436665807451790271994, −15.02352905207348476494708685692, −13.96344434221233479709604440300, −13.51965637546391069393821522362, −12.52710844694970278709892036550, −11.85283752213463725226009817409, −11.27034810975595988043517523254, −10.17859308422383559253818400717, −9.90578488818078727172430423297, −8.638168441768391057274801708592, −8.3442375617845873103752687231, −7.28862927783970022483943085523, −6.25868995751437090198543434281, −5.25484193056036452242130291745, −4.264782111963307306384429519717, −3.40881539947882151272231944108, −2.74292393405474705369616288548, −1.49947562806023087763976523545, −0.707425627302603113745591457927, 1.080142693102941835977625991396, 1.86286180326021136887295426859, 3.45855867362533503047450276188, 4.30512478993838043667031400236, 5.10928173925277097437579171756, 6.28664825403477832938578235250, 6.50888788088786857190416801658, 7.72804115482659168700258975352, 8.18532900568310019156591828480, 9.30805336417721613095511629419, 9.71491829694043969331671057925, 10.57393823508969886018845016451, 11.73846114425717426596827476470, 12.28719337570247757808160005970, 13.55177831124662198399509013369, 14.18407171523194293378337685482, 14.61863486887688283920707150556, 15.65061418207849421766351055279, 16.34535027122808659496889108788, 16.82979165509147110310594916854, 17.72567178754036502932760959492, 18.35075799853583259914124419327, 19.1285108810764501542828018429, 19.71606533826411115560817377254, 20.66471424287838772043657867932

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