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Properties

Label 2-63-7.4-c1-0-1
Degree $2$
Conductor $63$
Sign $0.968 + 0.250i$
Analytic cond. $0.503057$
Root an. cond. $0.709265$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)4-s + (−0.5 + 2.59i)7-s − 7·13-s + (−1.99 − 3.46i)16-s + (3.5 + 6.06i)19-s + (2.5 − 4.33i)25-s + (4 + 3.46i)28-s + (3.5 − 6.06i)31-s + (0.5 + 0.866i)37-s + 5·43-s + (−6.5 − 2.59i)49-s + (−7 + 12.1i)52-s + (−7 − 12.1i)61-s − 7.99·64-s + (−5.5 + 9.52i)67-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)4-s + (−0.188 + 0.981i)7-s − 1.94·13-s + (−0.499 − 0.866i)16-s + (0.802 + 1.39i)19-s + (0.5 − 0.866i)25-s + (0.755 + 0.654i)28-s + (0.628 − 1.08i)31-s + (0.0821 + 0.142i)37-s + 0.762·43-s + (−0.928 − 0.371i)49-s + (−0.970 + 1.68i)52-s + (−0.896 − 1.55i)61-s − 0.999·64-s + (−0.671 + 1.16i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.916115 - 0.116743i\)
\(L(\frac12)\) \(\approx\) \(0.916115 - 0.116743i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (0.5 - 2.59i)T \)
good2 \( 1 + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 7T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.89204274700472610334583207136, −14.19759421423389586027241204546, −12.43916690255645319183688183218, −11.73034383769830250715065081985, −10.20281678875964561378998763774, −9.438431329900591574938370924126, −7.70102303608271730101848054391, −6.22998923426805380952819476845, −5.06428004548084692695021768767, −2.45311402063529036694048344147, 2.95585781887631646524441704782, 4.71954527606832227016107159929, 6.95697683731745953258666267602, 7.56481020328747985759872945590, 9.250577143948027650400233365546, 10.57513541047567563154004230669, 11.76131590485914725834823505798, 12.75045934194220757550721874333, 13.81857004699226797923647639741, 15.10391512386323464488553043475

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