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Properties

Label 2-63-63.25-c1-0-0
Degree $2$
Conductor $63$
Sign $0.650 - 0.759i$
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  − 1.84·2-s + (1.39 + 1.02i)3-s + 1.39·4-s + (−0.667 + 1.15i)5-s + (−2.56 − 1.89i)6-s + (1.90 + 1.83i)7-s + 1.12·8-s + (0.880 + 2.86i)9-s + (1.22 − 2.12i)10-s + (−0.756 − 1.31i)11-s + (1.93 + 1.43i)12-s + (−2.58 − 4.48i)13-s + (−3.50 − 3.38i)14-s + (−2.11 + 0.923i)15-s − 4.84·16-s + (0.774 − 1.34i)17-s + ⋯
L(s)  = 1  − 1.30·2-s + (0.804 + 0.594i)3-s + 0.695·4-s + (−0.298 + 0.516i)5-s + (−1.04 − 0.773i)6-s + (0.719 + 0.694i)7-s + 0.396·8-s + (0.293 + 0.955i)9-s + (0.388 − 0.673i)10-s + (−0.228 − 0.395i)11-s + (0.558 + 0.413i)12-s + (−0.717 − 1.24i)13-s + (−0.936 − 0.904i)14-s + (−0.547 + 0.238i)15-s − 1.21·16-s + (0.187 − 0.325i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 - 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.650 - 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.650 - 0.759i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ 0.650 - 0.759i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.549842 + 0.253005i\)
\(L(\frac12)\) \(\approx\) \(0.549842 + 0.253005i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.39 - 1.02i)T \)
7 \( 1 + (-1.90 - 1.83i)T \)
good2 \( 1 + 1.84T + 2T^{2} \)
5 \( 1 + (0.667 - 1.15i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.756 + 1.31i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.58 + 4.48i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.774 + 1.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.25 + 2.16i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.68 + 6.37i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.0309 - 0.0536i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.84T + 31T^{2} \)
37 \( 1 + (0.281 + 0.487i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.51 - 7.81i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.09 + 8.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.51T + 47T^{2} \)
53 \( 1 + (-0.755 + 1.30i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.44T + 59T^{2} \)
61 \( 1 - 3.23T + 61T^{2} \)
67 \( 1 - 6.93T + 67T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 + (1.37 - 2.38i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 5.91T + 79T^{2} \)
83 \( 1 + (-2.80 + 4.85i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.703 - 1.21i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.09 - 10.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−15.17231050682039468217508293274, −14.48302249051473007914699838285, −12.96072639195858380648454392302, −11.14111292530845837853086165010, −10.41832908658101375019531703800, −9.186973190416328442481277782439, −8.299410165770485562196423534085, −7.40712131426821042647704228874, −4.95428750136026086516854026604, −2.71428091851818167901520909891, 1.63497820121561289417728828021, 4.38903082395480488332444143863, 7.12205118485207131154979071237, 7.84832251008363824147625580996, 8.896611708419601340685049413799, 9.838218860591598645081239447963, 11.26835806608296318818349971212, 12.58670396726696259045298368357, 13.82987996575696409946862119818, 14.76631953783583131862563459468

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