↑

Properties

Label 1-53-53.13-r0-0-0
Degree $1$
Conductor $53$
Sign $0.171 - 0.985i$
Analytic cond. $0.246130$
Root an. cond. $0.246130$
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  + (−0.970 + 0.239i)2-s + (0.568 − 0.822i)3-s + (0.885 − 0.464i)4-s + (−0.354 − 0.935i)5-s + (−0.354 + 0.935i)6-s + (−0.970 + 0.239i)7-s + (−0.748 + 0.663i)8-s + (−0.354 − 0.935i)9-s + (0.568 + 0.822i)10-s + (0.120 − 0.992i)11-s + (0.120 − 0.992i)12-s + (0.885 + 0.464i)13-s + (0.885 − 0.464i)14-s + (−0.970 − 0.239i)15-s + (0.568 − 0.822i)16-s + (−0.748 − 0.663i)17-s + ⋯
L(s)  = 1  + (−0.970 + 0.239i)2-s + (0.568 − 0.822i)3-s + (0.885 − 0.464i)4-s + (−0.354 − 0.935i)5-s + (−0.354 + 0.935i)6-s + (−0.970 + 0.239i)7-s + (−0.748 + 0.663i)8-s + (−0.354 − 0.935i)9-s + (0.568 + 0.822i)10-s + (0.120 − 0.992i)11-s + (0.120 − 0.992i)12-s + (0.885 + 0.464i)13-s + (0.885 − 0.464i)14-s + (−0.970 − 0.239i)15-s + (0.568 − 0.822i)16-s + (−0.748 − 0.663i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $0.171 - 0.985i$
Analytic conductor: \(0.246130\)
Root analytic conductor: \(0.246130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (0:\ ),\ 0.171 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4584111458 - 0.3855411476i\)
\(L(\frac12)\) \(\approx\) \(0.4584111458 - 0.3855411476i\)
\(L(1)\) \(\approx\) \(0.6603277647 - 0.2736257588i\)
\(L(1)\) \(\approx\) \(0.6603277647 - 0.2736257588i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad53 \( 1 \)
good2 \( 1 + (-0.970 + 0.239i)T \)
3 \( 1 + (0.568 - 0.822i)T \)
5 \( 1 + (-0.354 - 0.935i)T \)
7 \( 1 + (-0.970 + 0.239i)T \)
11 \( 1 + (0.120 - 0.992i)T \)
13 \( 1 + (0.885 + 0.464i)T \)
17 \( 1 + (-0.748 - 0.663i)T \)
19 \( 1 + (0.885 + 0.464i)T \)
23 \( 1 + T \)
29 \( 1 + (0.120 + 0.992i)T \)
31 \( 1 + (0.120 + 0.992i)T \)
37 \( 1 + (0.568 - 0.822i)T \)
41 \( 1 + (0.120 - 0.992i)T \)
43 \( 1 + (0.568 + 0.822i)T \)
47 \( 1 + (-0.354 + 0.935i)T \)
59 \( 1 + (-0.354 + 0.935i)T \)
61 \( 1 + (-0.748 + 0.663i)T \)
67 \( 1 + (0.885 - 0.464i)T \)
71 \( 1 + (0.568 + 0.822i)T \)
73 \( 1 + (-0.748 - 0.663i)T \)
79 \( 1 + (-0.970 - 0.239i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.748 - 0.663i)T \)
97 \( 1 + (-0.354 - 0.935i)T \)
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

−33.53630266584358241428856744504, −32.80858024563134655132446678994, −31.01452038962992232092586665400, −30.33092784462712380468014273952, −28.78014521910412601517980079516, −27.79836166518909801278883973082, −26.57676754543527598223806631812, −26.0522627896568437339526956761, −25.10997956553308594872136032164, −22.94349425862199865184892809968, −21.94138819600647449639184881456, −20.457824041575033619569128028143, −19.68539445321740408509419069741, −18.605117904289124905710930966667, −17.17501841648950684765159289437, −15.71573413623384921062985873736, −15.16114926577080639051632144064, −13.24061129486769584445109680408, −11.3364152539392508175488349991, −10.28255184509123747161029001088, −9.36915109324095355228482729386, −7.8619829959085016667857100680, −6.54253000392116623524428176737, −3.80987578278788155316205082119, −2.65620689612566299093193948568, 1.11081698469210527795116693373, 3.14027101591301006472440475049, 5.92947486659231480891004340385, 7.21379621054623759704300542090, 8.69052135214954083273961601408, 9.20004399806118723796927845973, 11.32482425386427506918533488470, 12.61021598066528761140793293117, 13.93150519974502030374808453876, 15.76336551315617749543709027204, 16.48866962373863486234666916587, 18.07973163930242807376111977867, 19.11458232359174789404761833327, 19.88066016905206034533287546544, 20.99370931466757136841606071833, 23.21424855346167145293747354710, 24.3722084893482015785402216439, 25.061543835469618368316838742211, 26.18327780294111787587392900204, 27.2858620473582338839129177471, 28.93346431811331408696297355069, 29.09613010657582601528235489572, 30.86735830973950466622556712852, 32.00876930707777158537072179776, 33.055895999288720516127010940444

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