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Properties

Label 1-53-53.37-r0-0-0
Degree $1$
Conductor $53$
Sign $0.638 - 0.769i$
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  + (−0.885 − 0.464i)2-s + (0.354 − 0.935i)3-s + (0.568 + 0.822i)4-s + (0.748 + 0.663i)5-s + (−0.748 + 0.663i)6-s + (0.885 + 0.464i)7-s + (−0.120 − 0.992i)8-s + (−0.748 − 0.663i)9-s + (−0.354 − 0.935i)10-s + (−0.970 + 0.239i)11-s + (0.970 − 0.239i)12-s + (0.568 − 0.822i)13-s + (−0.568 − 0.822i)14-s + (0.885 − 0.464i)15-s + (−0.354 + 0.935i)16-s + (0.120 − 0.992i)17-s + ⋯
L(s)  = 1  + (−0.885 − 0.464i)2-s + (0.354 − 0.935i)3-s + (0.568 + 0.822i)4-s + (0.748 + 0.663i)5-s + (−0.748 + 0.663i)6-s + (0.885 + 0.464i)7-s + (−0.120 − 0.992i)8-s + (−0.748 − 0.663i)9-s + (−0.354 − 0.935i)10-s + (−0.970 + 0.239i)11-s + (0.970 − 0.239i)12-s + (0.568 − 0.822i)13-s + (−0.568 − 0.822i)14-s + (0.885 − 0.464i)15-s + (−0.354 + 0.935i)16-s + (0.120 − 0.992i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $0.638 - 0.769i$
Analytic conductor: \(0.246130\)
Root analytic conductor: \(0.246130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (0:\ ),\ 0.638 - 0.769i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6751717317 - 0.3171806913i\)
\(L(\frac12)\) \(\approx\) \(0.6751717317 - 0.3171806913i\)
\(L(1)\) \(\approx\) \(0.8074786230 - 0.2786833846i\)
\(L(1)\) \(\approx\) \(0.8074786230 - 0.2786833846i\)

Euler product

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

−33.510373099483693562579588606224, −32.76623930386356777320483062582, −31.61378347001114607149686295240, −29.911990019369689458210976401760, −28.32640045256574104300946068510, −28.000415822025923418893236698105, −26.34923175444503647791278537529, −26.03082392017854394882958245934, −24.507594957951301625664872164499, −23.59054065252218306779615650608, −21.399632575392748004078176977854, −20.82643797492209727063865606310, −19.614974844909444042524642023768, −17.98370699542820550091874050644, −16.943885975253090115557639526364, −16.02524916773961176664997906560, −14.71339989778307035077831205894, −13.57675566512879763756051016687, −11.13955642590211705889149265215, −10.185556650775437637253408689891, −8.91193708940297361074936656199, −8.012437788556971148783767824401, −5.89230392206407293018070371421, −4.578021343006548344945218275408, −1.97694138598700808987613410927, 1.797115167367357641537886736347, 2.88732652138579749393967383648, 5.91788516367215522517468875876, 7.500307249122334116513057869873, 8.43267696024705431871735148367, 10.01680292604681089255195542429, 11.30260403569911593115412974650, 12.62733997865327194771321456613, 13.91272679276731490310244270206, 15.39217731448703244667467087603, 17.366365873376299646571295672356, 18.254816450879339817814737274567, 18.69076969157295362420589892497, 20.422241897395264721652725610988, 21.13792490077139838226901770809, 22.77325122059751847387167860967, 24.47025687083130668652428174217, 25.40027913356529520065050557327, 26.178456855474730251893981766624, 27.59655271012295971347699537745, 28.832220801662832126739010309538, 29.81473456053430345063814624520, 30.53468843869544921565198564021, 31.61221531202569828249224871891, 33.83456539844835002477077042922

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