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Properties

Label 2-504-9.4-c1-0-14
Degree $2$
Conductor $504$
Sign $0.767 + 0.641i$
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  + (1.62 − 0.593i)3-s + (1.50 − 2.60i)5-s + (0.5 + 0.866i)7-s + (2.29 − 1.93i)9-s + (2.74 + 4.75i)11-s + (−0.421 + 0.730i)13-s + (0.901 − 5.13i)15-s − 4.59·17-s − 3.80·19-s + (1.32 + 1.11i)21-s + (4.48 − 7.76i)23-s + (−2.02 − 3.51i)25-s + (2.58 − 4.50i)27-s + (−0.974 − 1.68i)29-s + (−1.68 + 2.92i)31-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)3-s + (0.672 − 1.16i)5-s + (0.188 + 0.327i)7-s + (0.765 − 0.643i)9-s + (0.826 + 1.43i)11-s + (−0.117 + 0.202i)13-s + (0.232 − 1.32i)15-s − 1.11·17-s − 0.872·19-s + (0.289 + 0.242i)21-s + (0.934 − 1.61i)23-s + (−0.405 − 0.702i)25-s + (0.497 − 0.867i)27-s + (−0.180 − 0.313i)29-s + (−0.303 + 0.525i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 + 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.767 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.767 + 0.641i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.767 + 0.641i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07747 - 0.754304i\)
\(L(\frac12)\) \(\approx\) \(2.07747 - 0.754304i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 + 0.593i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-1.50 + 2.60i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.74 - 4.75i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.421 - 0.730i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.59T + 17T^{2} \)
19 \( 1 + 3.80T + 19T^{2} \)
23 \( 1 + (-4.48 + 7.76i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.974 + 1.68i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.68 - 2.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.49T + 37T^{2} \)
41 \( 1 + (3.69 - 6.39i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.11 + 5.39i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.57 - 11.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.05T + 53T^{2} \)
59 \( 1 + (5.04 - 8.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.84 + 4.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.71 + 4.71i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.81T + 71T^{2} \)
73 \( 1 + 10.7T + 73T^{2} \)
79 \( 1 + (1.37 + 2.37i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.35 - 11.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.74T + 89T^{2} \)
97 \( 1 + (-3.83 - 6.64i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.63923802948109295236354206415, −9.579678497187928675428133367317, −8.929419471434531881216571933378, −8.518227134054895719792352465983, −7.13233313626021359081871275093, −6.41852065874969406112593906604, −4.85650915613442731517176628762, −4.24429089034814043332830755327, −2.39845450577920131086949259838, −1.53650488282231717535295505106, 1.91064441985195680134543752457, 3.13568112250650753300697059927, 3.90735701616222709071918798347, 5.43692008642918954426895388715, 6.61073947480230779753139652322, 7.28968341102102366240016542250, 8.588565597962926391064458600181, 9.117298881221408904230504622584, 10.22630363196643530689723352840, 10.82994189319055130808296970427

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