Properties

Label 1-53-53.4-r0-0-0
Degree $1$
Conductor $53$
Sign $0.882 - 0.470i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $0.882 - 0.470i$
Analytic conductor: \(0.246130\)
Root analytic conductor: \(0.246130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (0:\ ),\ 0.882 - 0.470i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.199217227 - 0.2993812323i\)
\(L(\frac12)\) \(\approx\) \(1.199217227 - 0.2993812323i\)
\(L(1)\) \(\approx\) \(1.362202343 - 0.2153928929i\)
\(L(1)\) \(\approx\) \(1.362202343 - 0.2153928929i\)

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

−33.30964135454156209514238827108, −32.38289819794719767249528231413, −31.4095623488340001525252008892, −29.873496254932106537640208731766, −29.16163907895263690901690808267, −28.08199695408682178576029926034, −26.46177425692389198448262338166, −25.55148924567610084082647456271, −23.842639796475013682934010286237, −22.7166495089418157984074189303, −22.00057784655521601536975544968, −21.229316020299937736664463154034, −19.668208491801698190835297617091, −18.32861916496212250055265380825, −16.44655861964787723725324500450, −15.65424341732953437647798836529, −14.37203409859159725187729544515, −13.1651593044075887049335843136, −11.476885592862138819462626245139, −10.731032696142924773120224025181, −9.38298886556597810025172238721, −6.534818445234280543039152314841, −5.926317436879339843338779501886, −4.050828116448655885928317760476, −2.84928929790310190048540534935, 1.947417523915129799159332918034, 4.190028896679972139766549405224, 5.76618671308885416579779143778, 6.67636616740593116084510016415, 8.26809913245971761559332164446, 10.4161758072823289560611664085, 12.14739646707015853751207070005, 12.88006313956942625167872391353, 13.6816967022771790098784849404, 15.58085408352626753077129258438, 16.70253007285421694282679640941, 17.65569458421013660768101635598, 19.551731192905117631552516651486, 20.53561128710350281456717459047, 22.049177287050537312183715496159, 23.08717635307994850384527967810, 23.870370866328296245021438467352, 25.13973072308717435871220811172, 25.72268938603783290490421351752, 28.169356292763632379664656025451, 28.900184742439328818538759032061, 29.97133927464191989320525286711, 30.94775082524789076324834199627, 32.307082259573893515557804406519, 33.06722883436535412161509001374

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