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Properties

Label 1-53-53.15-r0-0-0
Degree $1$
Conductor $53$
Sign $0.733 + 0.679i$
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  + (0.120 + 0.992i)2-s + (0.885 − 0.464i)3-s + (−0.970 + 0.239i)4-s + (0.568 − 0.822i)5-s + (0.568 + 0.822i)6-s + (0.120 + 0.992i)7-s + (−0.354 − 0.935i)8-s + (0.568 − 0.822i)9-s + (0.885 + 0.464i)10-s + (−0.748 + 0.663i)11-s + (−0.748 + 0.663i)12-s + (−0.970 − 0.239i)13-s + (−0.970 + 0.239i)14-s + (0.120 − 0.992i)15-s + (0.885 − 0.464i)16-s + (−0.354 + 0.935i)17-s + ⋯
L(s)  = 1  + (0.120 + 0.992i)2-s + (0.885 − 0.464i)3-s + (−0.970 + 0.239i)4-s + (0.568 − 0.822i)5-s + (0.568 + 0.822i)6-s + (0.120 + 0.992i)7-s + (−0.354 − 0.935i)8-s + (0.568 − 0.822i)9-s + (0.885 + 0.464i)10-s + (−0.748 + 0.663i)11-s + (−0.748 + 0.663i)12-s + (−0.970 − 0.239i)13-s + (−0.970 + 0.239i)14-s + (0.120 − 0.992i)15-s + (0.885 − 0.464i)16-s + (−0.354 + 0.935i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $0.733 + 0.679i$
Analytic conductor: \(0.246130\)
Root analytic conductor: \(0.246130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (0:\ ),\ 0.733 + 0.679i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.028709829 + 0.4035422736i\)
\(L(\frac12)\) \(\approx\) \(1.028709829 + 0.4035422736i\)
\(L(1)\) \(\approx\) \(1.178274954 + 0.3708812245i\)
\(L(1)\) \(\approx\) \(1.178274954 + 0.3708812245i\)

Euler product

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

−32.96978166591226918147971829297, −31.866539909441182169253463903182, −30.92609268865117498787431290998, −29.774938779301785940849242953009, −29.13175110450275229662744294934, −27.103765933562763333909583248614, −26.81611451379617130662331612863, −25.58482342161635887835544979138, −23.84006261150969809356712697659, −22.41761541348124477316923255297, −21.443628514288174200445958334765, −20.55192780107101836708706046422, −19.39079372551118391912664607402, −18.40765932151635915063675947476, −16.87935803653000516551793485427, −14.8863332290130666636186864247, −13.97702271741612507170963844282, −13.115395882449996549591773003056, −10.96958289723667604859542804622, −10.26605294055902510454247420484, −9.0480957625715783428366397747, −7.352683601044491277688625312459, −4.96634927982109192186307002778, −3.436348635634779601372937798029, −2.25866829525819337451149343166, 2.298692662269186980886006650519, 4.59064128322224267916782864084, 5.98028058190151663766365664924, 7.62072661584468205002941944878, 8.72406861804389279450748765053, 9.64549683297582146957000577548, 12.67012566673118879361026331450, 13.01953625874372671339950421712, 14.722683496609459016481751018789, 15.371531540637179826863246846705, 17.07041321544232115794923136275, 18.074314624760451127843028322919, 19.305287115262038797850376655283, 20.88572585216669030792421606038, 21.88926559669651908265407372211, 23.65905491771171936303570731715, 24.56969065284255710824561141990, 25.31271956802206308303427837368, 26.16978131851636821843719352248, 27.65283202725638697505337940483, 28.86384262017632603374418967505, 30.5219856323796449211417527630, 31.63679977461518982394333432180, 32.13347037499750092835235268315, 33.40200348762804981379666643131

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