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Properties

Label 1-63-63.5-r0-0-0
Degree $1$
Conductor $63$
Sign $0.458 + 0.888i$
Analytic cond. $0.292570$
Root an. cond. $0.292570$
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  − 2-s + 4-s + (−0.5 + 0.866i)5-s − 8-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + 16-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (−0.5 − 0.866i)26-s + ⋯
L(s)  = 1  − 2-s + 4-s + (−0.5 + 0.866i)5-s − 8-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + 16-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (−0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.458 + 0.888i$
Analytic conductor: \(0.292570\)
Root analytic conductor: \(0.292570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 63,\ (0:\ ),\ 0.458 + 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4722624905 + 0.2879181612i\)
\(L(\frac12)\) \(\approx\) \(0.4722624905 + 0.2879181612i\)
\(L(1)\) \(\approx\) \(0.6264705227 + 0.1775076719i\)
\(L(1)\) \(\approx\) \(0.6264705227 + 0.1775076719i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 - T \)
5 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−32.348389767951487965651177037580, −30.960516330588055534471511520703, −29.66140723547706013147690524677, −28.70052134249470711974507532175, −27.56220733340232363711397707485, −26.99777473581565589680639221522, −25.47804411837381771352576134618, −24.583606440630414087263244385632, −23.586629555898687790052754529255, −21.82749424679310449757636561482, −20.41313336591647898575667679404, −19.789645409345598958874823447835, −18.49629061241007603230508582387, −17.28998592480318541642863956424, −16.20481612063602189671492627577, −15.3668206484893829524551734526, −13.42004687626282907333356920480, −11.914258106320820458639753911992, −10.950710305859315204199262510971, −9.26973773814064345799413023018, −8.4423257005689284039551823718, −7.09346273455244644891520100717, −5.38343530558828032643880024350, −3.26625914828131546065596578326, −1.021868433106423018683967475746, 2.01179094807131049536492374547, 3.82284367681430757100270631608, 6.338906743264940479081324619510, 7.31757837008397566867465294600, 8.693969179525716072348530628329, 10.08361252591722005074480021744, 11.17405291012657473346355845129, 12.27884699359985944453373258674, 14.40198667408917660338810249020, 15.403027686789349129685868901614, 16.638905233507476246505138771877, 17.8995850132375643981373921650, 18.8695130815744034461952179646, 19.80108944387174388793267754181, 21.05137500457567133131723827942, 22.49726452049546578779807347072, 23.73602551869180088087973592066, 25.0748588898366832589406832310, 26.14785576387879367775443220671, 26.9157450658739731177379914369, 28.04806127920920320277717095469, 29.0302856779482058632778286305, 30.380225792957543495527868506586, 31.0004654370436864005547046355, 33.0077291396425388068532932186

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