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Properties

Label 1-53-53.25-r0-0-0
Degree $1$
Conductor $53$
Sign $-0.502 - 0.864i$
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  + (0.354 − 0.935i)2-s + (−0.120 − 0.992i)3-s + (−0.748 − 0.663i)4-s + (0.970 − 0.239i)5-s + (−0.970 − 0.239i)6-s + (−0.354 + 0.935i)7-s + (−0.885 + 0.464i)8-s + (−0.970 + 0.239i)9-s + (0.120 − 0.992i)10-s + (0.568 − 0.822i)11-s + (−0.568 + 0.822i)12-s + (−0.748 + 0.663i)13-s + (0.748 + 0.663i)14-s + (−0.354 − 0.935i)15-s + (0.120 + 0.992i)16-s + (0.885 + 0.464i)17-s + ⋯
L(s)  = 1  + (0.354 − 0.935i)2-s + (−0.120 − 0.992i)3-s + (−0.748 − 0.663i)4-s + (0.970 − 0.239i)5-s + (−0.970 − 0.239i)6-s + (−0.354 + 0.935i)7-s + (−0.885 + 0.464i)8-s + (−0.970 + 0.239i)9-s + (0.120 − 0.992i)10-s + (0.568 − 0.822i)11-s + (−0.568 + 0.822i)12-s + (−0.748 + 0.663i)13-s + (0.748 + 0.663i)14-s + (−0.354 − 0.935i)15-s + (0.120 + 0.992i)16-s + (0.885 + 0.464i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $-0.502 - 0.864i$
Analytic conductor: \(0.246130\)
Root analytic conductor: \(0.246130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (0:\ ),\ -0.502 - 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4942964076 - 0.8585990914i\)
\(L(\frac12)\) \(\approx\) \(0.4942964076 - 0.8585990914i\)
\(L(1)\) \(\approx\) \(0.8133416769 - 0.7752761395i\)
\(L(1)\) \(\approx\) \(0.8133416769 - 0.7752761395i\)

Euler product

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

−33.5690466842908764528978597488, −32.72068679895308000751795386298, −32.09686477223292375670052628643, −30.46644680163664179410804505774, −29.28663442264638188395562666336, −27.6755439078865654590571460940, −26.68238894604723656687977019349, −25.75146383500437438682557192686, −24.843604028920779935072194984567, −23.01757576099303267425030499320, −22.48025069305149336592311702594, −21.30405084722324793578895080028, −20.1088044690912189099508509835, −17.87843481501607267158272590693, −17.06153279856835297012959869397, −16.09322380539395313065681109499, −14.61519535182024487728919763363, −13.9577258005833326767507671423, −12.311401609437040617332018215550, −10.1445740632411993613359814304, −9.5167282833699589851414816214, −7.54369421658015629235940997436, −6.06994924182853995673295391877, −4.82616551079987322049069178809, −3.364494201405403378877903230574, 1.59086792655426903007158726037, 2.90284628429741877190039083080, 5.34027812958640334224853391224, 6.32592816031808680715932573425, 8.653494217623241520566964026462, 9.76873689789944420597201463295, 11.588134891999928449498507833407, 12.44101513902664073837304219754, 13.59440310530017895526567896281, 14.49579403393510004593269455662, 16.80051032042339450821259458419, 18.12596653311923533034341657850, 18.96351215139006536375939132505, 20.04485293876706227893470354243, 21.69007142795959756327677567039, 22.16985585030938049855571972711, 23.875400835595583669746341273461, 24.6727320096329733134687688833, 25.97968191425698754028868301535, 27.83929664416747383175947146360, 28.87848810390536285686267240302, 29.45778471874089473606031779075, 30.51173379889316199027408536748, 31.712454315860223450847072097756, 32.56261184463008714032850764383

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