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Properties

Label 2-504-63.4-c1-0-9
Degree $2$
Conductor $504$
Sign $0.592 + 0.805i$
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  + (−0.341 − 1.69i)3-s + 0.526·5-s + (2.43 + 1.02i)7-s + (−2.76 + 1.16i)9-s + 4.61·11-s + (0.244 − 0.423i)13-s + (−0.179 − 0.893i)15-s + (2.75 − 4.77i)17-s + (1.83 + 3.18i)19-s + (0.904 − 4.49i)21-s − 0.0539·23-s − 4.72·25-s + (2.91 + 4.30i)27-s + (−3.28 − 5.68i)29-s + (−3.03 − 5.26i)31-s + ⋯
L(s)  = 1  + (−0.197 − 0.980i)3-s + 0.235·5-s + (0.922 + 0.386i)7-s + (−0.922 + 0.386i)9-s + 1.39·11-s + (0.0678 − 0.117i)13-s + (−0.0464 − 0.230i)15-s + (0.668 − 1.15i)17-s + (0.421 + 0.730i)19-s + (0.197 − 0.980i)21-s − 0.0112·23-s − 0.944·25-s + (0.561 + 0.827i)27-s + (−0.609 − 1.05i)29-s + (−0.545 − 0.945i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.592 + 0.805i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.592 + 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41501 - 0.715389i\)
\(L(\frac12)\) \(\approx\) \(1.41501 - 0.715389i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.341 + 1.69i)T \)
7 \( 1 + (-2.43 - 1.02i)T \)
good5 \( 1 - 0.526T + 5T^{2} \)
11 \( 1 - 4.61T + 11T^{2} \)
13 \( 1 + (-0.244 + 0.423i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.75 + 4.77i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.83 - 3.18i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.0539T + 23T^{2} \)
29 \( 1 + (3.28 + 5.68i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.03 + 5.26i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.223 - 0.387i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.52 + 4.36i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.84 - 4.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.59 + 7.96i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.37 - 7.57i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.31 - 5.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.232 + 0.403i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.59 + 4.49i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.76T + 71T^{2} \)
73 \( 1 + (5.23 - 9.07i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.18 - 14.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.49 - 7.78i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.05 + 12.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.22 - 9.04i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.16662096704495971067276262513, −9.759662846625592551033798324745, −8.960397646362298184144992677355, −7.88788368929033202230037570823, −7.29279342088072544987027699877, −6.04525190203253694700172631995, −5.45325534494448147704357868536, −3.98802084907591521889385300040, −2.36968825754051751187730441321, −1.21460519973619092970711817697, 1.52849624318242406249539035115, 3.46758491578798000542711925492, 4.29091438846897907825908674306, 5.32072959736495867300669139320, 6.26853769487439721236942643740, 7.46659271068569345428886315072, 8.630724672660154862552052990931, 9.282681184215532341123264653769, 10.22341040583436892025669114809, 11.04094865224842561970373375196

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