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Properties

Label 1-63-63.32-r1-0-0
Degree $1$
Conductor $63$
Sign $-0.805 - 0.592i$
Analytic cond. $6.77029$
Root an. cond. $6.77029$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s − 5-s − 8-s + (−0.5 − 0.866i)10-s − 11-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)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 − 23-s + 25-s + (0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s − 5-s − 8-s + (−0.5 − 0.866i)10-s − 11-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)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 − 23-s + 25-s + (0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.805 - 0.592i$
Analytic conductor: \(6.77029\)
Root analytic conductor: \(6.77029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 63,\ (1:\ ),\ -0.805 - 0.592i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1523427185 + 0.4637776184i\)
\(L(\frac12)\) \(\approx\) \(-0.1523427185 + 0.4637776184i\)
\(L(1)\) \(\approx\) \(0.6374146630 + 0.4694082409i\)
\(L(1)\) \(\approx\) \(0.6374146630 + 0.4694082409i\)

Euler product

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

−31.50430152621943434703189349017, −30.42010497109819597981691435296, −29.28868487745808144757263928577, −28.22359352980773594430243947320, −27.273036213713512915670246896984, −26.13825319004199676647898111630, −24.15262341526251445917825537758, −23.550537789947449390756680071386, −22.39247429235318845127978095058, −21.221727188370206708627332744028, −20.11116834892003220421090614301, −19.144332429991054335629986971517, −18.14794544565383608713697534769, −16.22068045351388620290112126219, −15.0667842415723153246067225999, −13.821245985967779568047736776893, −12.464404210585805112806457952090, −11.543492680274314995541217001905, −10.36209178411306790975518471829, −8.86147525634911443535607368494, −7.207873796731088656840733616976, −5.22726535841865591825911967903, −3.997194976488549994666400337440, −2.47227280735772256264480903533, −0.21101117689275705024907535069, 3.17955636089342400003914033975, 4.56365513714701777682869607341, 5.97673082023526117967730544775, 7.65923530927053757782151187933, 8.26901164242919121145894060990, 10.31351580554874013154419967473, 12.072478763352504260770963856581, 12.94746660762414333181602887927, 14.53234408657870888604843767065, 15.45918312962319185372878980082, 16.40609984630526446451048265662, 17.73561260779734798290115972659, 19.01584233836184106453203757147, 20.48038571443826156997941289305, 21.76339284304054345972240506932, 23.05074015715147677869480646583, 23.66250953355354062153317323445, 24.8330744310206676912662106752, 26.04506879730947531674930544654, 27.01013465197015233988404774550, 28.032459431641616801888324061593, 29.78537853009936930716878815426, 30.86482576814273861475276657861, 31.77382675878824807691733348230, 32.5609969293651311198500232679

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