Properties

Label 2-63-63.4-c1-0-4
Degree $2$
Conductor $63$
Sign $0.455 + 0.890i$
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  + (1.19 − 2.06i)2-s + (0.266 + 1.71i)3-s + (−1.84 − 3.20i)4-s − 2.92·5-s + (3.85 + 1.49i)6-s + (2.35 + 1.20i)7-s − 4.05·8-s + (−2.85 + 0.913i)9-s + (−3.48 + 6.03i)10-s − 1.35·11-s + (4.98 − 4.01i)12-s + (−0.733 + 1.26i)13-s + (5.29 − 3.43i)14-s + (−0.779 − 4.99i)15-s + (−1.13 + 1.96i)16-s + (1.65 − 2.86i)17-s + ⋯
L(s)  = 1  + (0.843 − 1.46i)2-s + (0.154 + 0.988i)3-s + (−0.924 − 1.60i)4-s − 1.30·5-s + (1.57 + 0.608i)6-s + (0.891 + 0.453i)7-s − 1.43·8-s + (−0.952 + 0.304i)9-s + (−1.10 + 1.90i)10-s − 0.408·11-s + (1.43 − 1.16i)12-s + (−0.203 + 0.352i)13-s + (1.41 − 0.919i)14-s + (−0.201 − 1.29i)15-s + (−0.284 + 0.492i)16-s + (0.401 − 0.695i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.455 + 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.455 + 0.890i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ 0.455 + 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.982468 - 0.600807i\)
\(L(\frac12)\) \(\approx\) \(0.982468 - 0.600807i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.266 - 1.71i)T \)
7 \( 1 + (-2.35 - 1.20i)T \)
good2 \( 1 + (-1.19 + 2.06i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 2.92T + 5T^{2} \)
11 \( 1 + 1.35T + 11T^{2} \)
13 \( 1 + (0.733 - 1.26i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.65 + 2.86i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.10 + 1.91i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.62T + 23T^{2} \)
29 \( 1 + (-0.521 - 0.903i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.63 + 2.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.43 - 9.41i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.904 - 1.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.17 + 3.76i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.98 - 3.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.22 - 5.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.10 - 10.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.279 - 0.484i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.40 + 11.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + (-5.22 + 9.05i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.383 - 0.664i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.983 + 1.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.20 - 5.54i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.14 + 7.17i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.78067739915754840144998284638, −13.63233750479999278852533229178, −12.11722546718272943896380855097, −11.47210459514479716300383983776, −10.74637762699573106940765325429, −9.370663155108345699689735412055, −7.978321057653895087580353550234, −5.11397260084066045235725927060, −4.28790500631972028391276575958, −2.89378383945495108507831119830, 3.84184665474636678764671848242, 5.35404388869163689800683830451, 6.90715392699090883111506020060, 7.85854727084343000468233115495, 8.279779662982566325496196472502, 11.08241246194059462622958042965, 12.32732877707333912346962592835, 13.14529526093128969126645940228, 14.43285985099618476812732854176, 14.87948190725450317951336938574

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