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Properties

Label 1-53-53.49-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

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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: $\chi_{53} (49, \cdot )$
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 \( 1 + (-0.354 + 0.935i)T \)
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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

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

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