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Properties

Label 2-504-63.20-c1-0-13
Degree $2$
Conductor $504$
Sign $0.872 - 0.488i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
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  + (1.69 − 0.346i)3-s + (1.79 + 3.10i)5-s + (−2.56 − 0.634i)7-s + (2.75 − 1.17i)9-s + (−0.200 − 0.115i)11-s + (1.16 − 0.673i)13-s + (4.11 + 4.64i)15-s + 7.94·17-s + 3.06i·19-s + (−4.57 − 0.185i)21-s + (−4.87 + 2.81i)23-s + (−3.91 + 6.78i)25-s + (4.27 − 2.95i)27-s + (−2.33 − 1.34i)29-s + (1.85 − 1.07i)31-s + ⋯
L(s)  = 1  + (0.979 − 0.200i)3-s + (0.801 + 1.38i)5-s + (−0.970 − 0.239i)7-s + (0.919 − 0.392i)9-s + (−0.0604 − 0.0348i)11-s + (0.323 − 0.186i)13-s + (1.06 + 1.19i)15-s + 1.92·17-s + 0.702i·19-s + (−0.999 − 0.0403i)21-s + (−1.01 + 0.586i)23-s + (−0.783 + 1.35i)25-s + (0.822 − 0.568i)27-s + (−0.433 − 0.250i)29-s + (0.332 − 0.192i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.872 - 0.488i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.872 - 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04487 + 0.533407i\)
\(L(\frac12)\) \(\approx\) \(2.04487 + 0.533407i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.69 + 0.346i)T \)
7 \( 1 + (2.56 + 0.634i)T \)
good5 \( 1 + (-1.79 - 3.10i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.200 + 0.115i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.16 + 0.673i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 7.94T + 17T^{2} \)
19 \( 1 - 3.06iT - 19T^{2} \)
23 \( 1 + (4.87 - 2.81i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.33 + 1.34i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.85 + 1.07i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.27T + 37T^{2} \)
41 \( 1 + (0.813 + 1.40i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.927 - 1.60i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.0396 - 0.0686i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 11.6iT - 53T^{2} \)
59 \( 1 + (6.48 + 11.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.729 - 0.420i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.05 + 8.75i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.47iT - 71T^{2} \)
73 \( 1 - 7.97iT - 73T^{2} \)
79 \( 1 + (3.30 - 5.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.41 + 11.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.56T + 89T^{2} \)
97 \( 1 + (13.1 + 7.58i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.60048739484787449167520964380, −9.926490878861464010851191977618, −9.598043691744792247630780953932, −8.145945328213616117527967819281, −7.37505996133296488817948256830, −6.47766651394079791138298026822, −5.70140920607845125315159442241, −3.58594895883412851137395692877, −3.19560636534850572322270019872, −1.85718735129666186894263518830, 1.38301720439243071305615472362, 2.80109073370153259809820439919, 3.99282004226477122740161483082, 5.17364809334650547575232398142, 6.06645309514888396167124364819, 7.38940561860672568288025127758, 8.478607282540438618455005025964, 9.055882936174746190416819722812, 9.797641232303216969301497560168, 10.36761351810986428853908051081

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