↑

Properties

Label 1-245-245.149-r0-0-0
Degree $1$
Conductor $245$
Sign $0.999 + 0.0427i$
Analytic cond. $1.13777$
Root an. cond. $1.13777$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.826 − 0.563i)2-s + (0.733 + 0.680i)3-s + (0.365 + 0.930i)4-s + (−0.222 − 0.974i)6-s + (0.222 − 0.974i)8-s + (0.0747 + 0.997i)9-s + (0.0747 − 0.997i)11-s + (−0.365 + 0.930i)12-s + (0.900 − 0.433i)13-s + (−0.733 + 0.680i)16-s + (0.988 − 0.149i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + (−0.623 + 0.781i)22-s + (0.988 + 0.149i)23-s + (0.826 − 0.563i)24-s + ⋯
L(s)  = 1  + (−0.826 − 0.563i)2-s + (0.733 + 0.680i)3-s + (0.365 + 0.930i)4-s + (−0.222 − 0.974i)6-s + (0.222 − 0.974i)8-s + (0.0747 + 0.997i)9-s + (0.0747 − 0.997i)11-s + (−0.365 + 0.930i)12-s + (0.900 − 0.433i)13-s + (−0.733 + 0.680i)16-s + (0.988 − 0.149i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + (−0.623 + 0.781i)22-s + (0.988 + 0.149i)23-s + (0.826 − 0.563i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.999 + 0.0427i$
Analytic conductor: \(1.13777\)
Root analytic conductor: \(1.13777\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 245,\ (0:\ ),\ 0.999 + 0.0427i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.108039485 + 0.02368393723i\)
\(L(\frac12)\) \(\approx\) \(1.108039485 + 0.02368393723i\)
\(L(1)\) \(\approx\) \(0.9797183734 + 0.01396592123i\)
\(L(1)\) \(\approx\) \(0.9797183734 + 0.01396592123i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.826 - 0.563i)T \)
3 \( 1 + (0.733 + 0.680i)T \)
11 \( 1 + (0.0747 - 0.997i)T \)
13 \( 1 + (0.900 - 0.433i)T \)
17 \( 1 + (0.988 - 0.149i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.988 + 0.149i)T \)
29 \( 1 + (0.623 + 0.781i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.365 + 0.930i)T \)
41 \( 1 + (-0.222 + 0.974i)T \)
43 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (-0.826 - 0.563i)T \)
53 \( 1 + (-0.365 - 0.930i)T \)
59 \( 1 + (0.955 + 0.294i)T \)
61 \( 1 + (0.365 - 0.930i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (-0.826 + 0.563i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.900 + 0.433i)T \)
89 \( 1 + (0.0747 + 0.997i)T \)
97 \( 1 - T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.76554756411322251914511303035, −25.465687639040200839989985689234, −24.5572032283174672826952392716, −23.484711139343427829788704497465, −22.98369425667215094279490115275, −20.94706871769816596199346661939, −20.49971143991754748461711558288, −19.189068457316160704702620226354, −18.82029714312672224780729150407, −17.78325784552448126650420115751, −16.93115468892636983245884967158, −15.73420824244979597725584823865, −14.79335943396600963108081158024, −14.10975673685594509549171316460, −12.87559068049984930126649543373, −11.78459039298217244072348869880, −10.41309633885963861413458644718, −9.41302172668898515585375844822, −8.52051423783226507039408979999, −7.602803179037900806744278390862, −6.75183691036598361238010893834, −5.693633134222819867330705817189, −3.95779405994059734854560605011, −2.2957129494710251029688089536, −1.25404014216734083299977629085, 1.29467438636547355950666762749, 2.963138075041238668675034466001, 3.489039156583410756400107548626, 5.02746017050185980967308446283, 6.71266951782976046245606337738, 8.121749529957802740625815667, 8.661725686236693046949760213253, 9.65911613817292776836246465786, 10.69363752580319496844291972541, 11.32857381036014156350564786937, 12.8359376473179704542767539067, 13.69974912767948057794123689120, 14.94803133183892868987098336052, 16.05907872289547657442047043073, 16.614078981182691547374487518341, 17.88411179529849563104601269868, 18.93950205353060150492914682347, 19.57790871415736000255466646734, 20.56154518634150898639113613780, 21.31996642875817115956651611790, 21.92330860893041265876951720019, 23.27345341450236634684785574669, 24.72634234104884579455339216595, 25.54049004827356178891451563175, 26.16504410181580113189885018129

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