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Properties

Label 2-63-21.20-c1-0-0
Degree $2$
Conductor $63$
Sign $-0.577 - 0.816i$
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  + 2.57i·2-s − 4.64·4-s + 2.64·7-s − 6.82i·8-s − 0.913i·11-s + 6.82i·14-s + 8.29·16-s + 2.35·22-s − 9.39i·23-s − 5·25-s − 12.2·28-s + 6.06i·29-s + 7.73i·32-s − 10.5·37-s + 5.29·43-s + 4.24i·44-s + ⋯
L(s)  = 1  + 1.82i·2-s − 2.32·4-s + 0.999·7-s − 2.41i·8-s − 0.275i·11-s + 1.82i·14-s + 2.07·16-s + 0.501·22-s − 1.95i·23-s − 25-s − 2.32·28-s + 1.12i·29-s + 1.36i·32-s − 1.73·37-s + 0.806·43-s + 0.639i·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ -0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.386282 + 0.746240i\)
\(L(\frac12)\) \(\approx\) \(0.386282 + 0.746240i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.64T \)
good2 \( 1 - 2.57iT - 2T^{2} \)
5 \( 1 + 5T^{2} \)
11 \( 1 + 0.913iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 9.39iT - 23T^{2} \)
29 \( 1 - 6.06iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 10.5T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 5.29T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 14.5iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 7.57iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−15.37524960410415049491509886586, −14.43840521773570063701630613924, −13.79772829381197035089787146869, −12.35861143793256988617787627933, −10.63294848724719968717501180714, −8.964968723886769494525121918646, −8.127330308480759372918817842344, −6.97782560947259423579006266016, −5.64820847737513046977322495616, −4.43966950728793303655486985089, 1.88132894244762286071273836156, 3.79668426323100196691206950748, 5.21610005166815684542841725562, 7.86102276266248389000501685141, 9.209824168885707020080376475866, 10.25628476622249935753522906366, 11.39760454038416226698064319009, 11.97825212513289685801867020237, 13.30674372532427752948356440329, 14.09350761082491907877145856572

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