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Properties

Label 1-1575-1575.1244-r0-0-0
Degree $1$
Conductor $1575$
Sign $0.564 - 0.825i$
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  + (−0.104 − 0.994i)2-s + (−0.978 + 0.207i)4-s + (0.309 + 0.951i)8-s + (0.809 − 0.587i)11-s + (0.913 − 0.406i)13-s + (0.913 − 0.406i)16-s + (0.978 + 0.207i)17-s + (−0.669 + 0.743i)19-s + (−0.669 − 0.743i)22-s + (−0.809 + 0.587i)23-s + (−0.5 − 0.866i)26-s + (0.978 − 0.207i)29-s + (−0.669 + 0.743i)31-s + (−0.5 − 0.866i)32-s + (0.104 − 0.994i)34-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (−0.978 + 0.207i)4-s + (0.309 + 0.951i)8-s + (0.809 − 0.587i)11-s + (0.913 − 0.406i)13-s + (0.913 − 0.406i)16-s + (0.978 + 0.207i)17-s + (−0.669 + 0.743i)19-s + (−0.669 − 0.743i)22-s + (−0.809 + 0.587i)23-s + (−0.5 − 0.866i)26-s + (0.978 − 0.207i)29-s + (−0.669 + 0.743i)31-s + (−0.5 − 0.866i)32-s + (0.104 − 0.994i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.564 - 0.825i$
Analytic conductor: \(7.31426\)
Root analytic conductor: \(7.31426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1575,\ (0:\ ),\ 0.564 - 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.305469609 - 0.6892028073i\)
\(L(\frac12)\) \(\approx\) \(1.305469609 - 0.6892028073i\)
\(L(1)\) \(\approx\) \(0.9449166719 - 0.4302054757i\)
\(L(1)\) \(\approx\) \(0.9449166719 - 0.4302054757i\)

Euler product

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

−20.63747573386342400908085532455, −19.71679129829190919209715006253, −18.95738833640602861681162789535, −18.266867683614213396840390977636, −17.54662359898233408466869343905, −16.823028047963813280319791424231, −16.17642447303402044786791583888, −15.46037417110821619853193480810, −14.57009481857892677931276104434, −14.146567408861254348644851177395, −13.23364039234279136207508674, −12.46673648851663374236206390994, −11.58407075547992697704891911150, −10.52945308276590431987679611640, −9.694993715880651932071993873178, −9.00451961894608755655554161354, −8.24614118379899899046491546849, −7.436312957794389372598945540381, −6.493437873459106851054094689656, −6.10508195232695110545490376975, −4.90987800528938482195494411593, −4.2438956273512336338102226457, −3.379303489377405023783522564363, −1.90675392553872981048454292211, −0.79481917076706193214865223109, 0.93105665945399073415165004319, 1.65116417590115508474883738929, 2.84240665708279093621435438058, 3.71356568418523004533583960927, 4.200054159526699914957156908719, 5.58754889208271599091799374591, 6.04646692786152505116088386038, 7.449330596803312545518425884824, 8.34864980409563424314820345833, 8.87407382894795909359407850717, 9.85504163973805034203426144003, 10.53526089586177698472239040102, 11.23450739571270357976683790730, 12.081614520581192580139198267735, 12.6154267600154118386622674836, 13.591114140710696555946247647363, 14.182599236398069111471737142552, 14.88617048173311592170446672231, 16.181747189525018800183767079934, 16.66689512350965047514074658493, 17.75828263779142148746140607505, 18.13665249690708205543411645158, 19.22719876536903207098486337150, 19.48417651187381243241584746724, 20.38837804396000360302141535123

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