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Properties

Label 1-53-53.34-r1-0-0
Degree $1$
Conductor $53$
Sign $0.553 + 0.833i$
Analytic cond. $5.69564$
Root an. cond. $5.69564$
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  + (0.239 + 0.970i)2-s + (−0.822 − 0.568i)3-s + (−0.885 + 0.464i)4-s + (0.935 − 0.354i)5-s + (0.354 − 0.935i)6-s + (0.970 − 0.239i)7-s + (−0.663 − 0.748i)8-s + (0.354 + 0.935i)9-s + (0.568 + 0.822i)10-s + (−0.120 + 0.992i)11-s + (0.992 + 0.120i)12-s + (0.885 + 0.464i)13-s + (0.464 + 0.885i)14-s + (−0.970 − 0.239i)15-s + (0.568 − 0.822i)16-s + (0.748 + 0.663i)17-s + ⋯
L(s)  = 1  + (0.239 + 0.970i)2-s + (−0.822 − 0.568i)3-s + (−0.885 + 0.464i)4-s + (0.935 − 0.354i)5-s + (0.354 − 0.935i)6-s + (0.970 − 0.239i)7-s + (−0.663 − 0.748i)8-s + (0.354 + 0.935i)9-s + (0.568 + 0.822i)10-s + (−0.120 + 0.992i)11-s + (0.992 + 0.120i)12-s + (0.885 + 0.464i)13-s + (0.464 + 0.885i)14-s + (−0.970 − 0.239i)15-s + (0.568 − 0.822i)16-s + (0.748 + 0.663i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.553 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.553 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $0.553 + 0.833i$
Analytic conductor: \(5.69564\)
Root analytic conductor: \(5.69564\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (1:\ ),\ 0.553 + 0.833i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.398560573 + 0.7501349830i\)
\(L(\frac12)\) \(\approx\) \(1.398560573 + 0.7501349830i\)
\(L(1)\) \(\approx\) \(1.090759177 + 0.4053417945i\)
\(L(1)\) \(\approx\) \(1.090759177 + 0.4053417945i\)

Euler product

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

−32.91231128883179472603724916890, −31.784576983038491792436657611342, −30.286735258768792959915058315766, −29.4916506569730252538842023592, −28.482948514530618004847165343255, −27.48090975656765218575145811758, −26.5461071020070558405387409475, −24.66300850673611152615687616145, −23.2595327444424705059061441090, −22.26385734431450532880970499032, −21.20331756903097404158551366063, −20.73838340860255807829048362772, −18.48435611115395443761293641375, −18.01978191374784636148654167675, −16.55059948064708577237297470635, −14.74041893653025115070918738764, −13.68228936155280571313421794058, −12.0843170029523656104740108970, −10.9556109890602129120740152741, −10.168166043398435789246541886955, −8.678785342995819310064487266501, −5.95607712933566577830170512500, −5.08246361335702032640867124389, −3.261331892954154823875513271152, −1.21994382139713066928852310299, 1.44809098712415726253580982193, 4.6309877172181488878369887792, 5.65406282516164545260130057284, 6.92530285809496626496719740886, 8.24600665569764319278223331772, 9.95587358768575709386983605357, 11.75744007193267015942985928047, 13.149243160522752551630059547666, 13.98593796155090179094590997585, 15.57906507224658092649513815504, 17.151584940513792919363116201896, 17.50051314275075842440379889700, 18.62551720062713958897929827158, 20.87295461584376817575197906610, 21.916211972769956501977173690441, 23.26948069515502870651141613557, 24.01514807141779943296253049438, 25.057164586170032449943043552173, 26.04777150545499059747345866946, 27.74792287281963803306730704109, 28.48525074697870423265404110159, 30.15265392442899208850398115516, 30.82459814742208857890742216664, 32.55623809369897480580419321077, 33.55207663200416916636215093603

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