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Properties

Label 1-53-53.28-r0-0-0
Degree $1$
Conductor $53$
Sign $-0.984 + 0.178i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $-0.984 + 0.178i$
Analytic conductor: \(0.246130\)
Root analytic conductor: \(0.246130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (0:\ ),\ -0.984 + 0.178i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04793650305 + 0.5341174634i\)
\(L(\frac12)\) \(\approx\) \(0.04793650305 + 0.5341174634i\)
\(L(1)\) \(\approx\) \(0.4103233777 + 0.5356937286i\)
\(L(1)\) \(\approx\) \(0.4103233777 + 0.5356937286i\)

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

−32.356730873732396335413652152679, −31.37971125856289786100772826696, −30.23627499831025673172693543506, −29.73009789009736815081312595247, −28.36246937753707088193080983767, −27.3404094712552959907391160531, −26.17473106296294412902071610843, −24.84380132004471656173943280206, −23.20394610716118066878050561537, −22.81154123281715177783120265516, −20.71647959172912549616448966043, −19.67835827860106556417698099412, −19.27467778162323373946340789163, −17.68384557900344202405591162766, −16.787911937272009441520070129499, −14.66865040062148031163433508733, −13.13696109623428329461697931472, −12.3298160148038102551313103368, −11.20537739655518055665214004279, −9.601375862849303974736530033211, −8.006487739365523644353784272302, −7.135576567471251730157398644961, −4.45980338516380275499274853275, −2.8945212949372291124191755653, −0.84217888060740760289442821386, 3.47380813115694702484227011387, 4.97389922988203504101127431329, 6.46073193479141728574752200087, 8.24566400431848730295491620586, 9.138998607071035207274173872, 10.56512632198935435676054538167, 12.10438758185552558109404251940, 14.33287076781283010941614658914, 15.072232199124506923612445427277, 16.14598236592983130356414115307, 16.949682075614479574116117315679, 18.91426243579941739592285886905, 19.462070745866109053845198447119, 21.474195493604732611465176268457, 22.46459424052235913053021607127, 23.571693861434072392880791144718, 24.949835046407358178993997817553, 26.02191523879944375393910764985, 27.16742247658275048375441595071, 27.62317598587406516313856462294, 28.93318619060869433623807779619, 31.1268809826543289560989372612, 31.88418971816748550817534538992, 32.6742984733135576625898578875, 34.27822130153401011454487176945

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