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Properties

Label 2-507-13.12-c3-0-64
Degree $2$
Conductor $507$
Sign $-0.832 - 0.554i$
Analytic cond. $29.9139$
Root an. cond. $5.46936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.36i·2-s − 3·3-s + 2.42·4-s − 6.42i·5-s + 7.08i·6-s − 29.4i·7-s − 24.6i·8-s + 9·9-s − 15.1·10-s − 0.624i·11-s − 7.26·12-s − 69.6·14-s + 19.2i·15-s − 38.7·16-s − 87.7·17-s − 21.2i·18-s + ⋯
L(s)  = 1  − 0.835i·2-s − 0.577·3-s + 0.302·4-s − 0.574i·5-s + 0.482i·6-s − 1.59i·7-s − 1.08i·8-s + 0.333·9-s − 0.479·10-s − 0.0171i·11-s − 0.174·12-s − 1.32·14-s + 0.331i·15-s − 0.605·16-s − 1.25·17-s − 0.278i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.832 - 0.554i$
Analytic conductor: \(29.9139\)
Root analytic conductor: \(5.46936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :3/2),\ -0.832 - 0.554i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.420788419\)
\(L(\frac12)\) \(\approx\) \(1.420788419\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
13 \( 1 \)
good2 \( 1 + 2.36iT - 8T^{2} \)
5 \( 1 + 6.42iT - 125T^{2} \)
7 \( 1 + 29.4iT - 343T^{2} \)
11 \( 1 + 0.624iT - 1.33e3T^{2} \)
17 \( 1 + 87.7T + 4.91e3T^{2} \)
19 \( 1 + 82.8iT - 6.85e3T^{2} \)
23 \( 1 - 74.7T + 1.21e4T^{2} \)
29 \( 1 - 226.T + 2.43e4T^{2} \)
31 \( 1 + 173. iT - 2.97e4T^{2} \)
37 \( 1 - 112. iT - 5.06e4T^{2} \)
41 \( 1 - 267. iT - 6.89e4T^{2} \)
43 \( 1 + 383.T + 7.95e4T^{2} \)
47 \( 1 - 337. iT - 1.03e5T^{2} \)
53 \( 1 + 146.T + 1.48e5T^{2} \)
59 \( 1 + 529. iT - 2.05e5T^{2} \)
61 \( 1 - 203.T + 2.26e5T^{2} \)
67 \( 1 + 121. iT - 3.00e5T^{2} \)
71 \( 1 - 661. iT - 3.57e5T^{2} \)
73 \( 1 - 167. iT - 3.89e5T^{2} \)
79 \( 1 + 101.T + 4.93e5T^{2} \)
83 \( 1 - 506. iT - 5.71e5T^{2} \)
89 \( 1 - 1.40e3iT - 7.04e5T^{2} \)
97 \( 1 + 1.90e3iT - 9.12e5T^{2} \)
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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

−10.21303476544917445149041984838, −9.455699323674051821793372627342, −8.183296316762252669304829154263, −6.90834972619760966749840351621, −6.59948692652107903236876183569, −4.82670184589535679534561360524, −4.20653752096597986898700392987, −2.86661311240178750697275406180, −1.30796239569681996513018041787, −0.48404589162448832065162214192, 1.95599148723013546685785741575, 2.99256078026543434940603333478, 4.85309214164114946846572829615, 5.65570862465279487880568299208, 6.48353257674666506677488389240, 7.04433889258397393472362311157, 8.368556092173329003990861706507, 8.916057819636261324252474124810, 10.31511188072694432694082885541, 11.04654030210883174719100095974

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