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Properties

Label 2-507-13.12-c3-0-0
Degree $2$
Conductor $507$
Sign $0.246 + 0.969i$
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  + 2.73i·2-s − 3·3-s + 0.524·4-s + 21.1i·5-s − 8.20i·6-s − 25.8i·7-s + 23.3i·8-s + 9·9-s − 57.7·10-s − 6.96i·11-s − 1.57·12-s + 70.7·14-s − 63.3i·15-s − 59.5·16-s − 122.·17-s + 24.6i·18-s + ⋯
L(s)  = 1  + 0.966i·2-s − 0.577·3-s + 0.0655·4-s + 1.88i·5-s − 0.558i·6-s − 1.39i·7-s + 1.03i·8-s + 0.333·9-s − 1.82·10-s − 0.191i·11-s − 0.0378·12-s + 1.34·14-s − 1.09i·15-s − 0.930·16-s − 1.75·17-s + 0.322i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.246 + 0.969i$
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: \((2,\ 507,\ (\ :3/2),\ 0.246 + 0.969i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.09124594594\)
\(L(\frac12)\) \(\approx\) \(0.09124594594\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.73iT - 8T^{2} \)
5 \( 1 - 21.1iT - 125T^{2} \)
7 \( 1 + 25.8iT - 343T^{2} \)
11 \( 1 + 6.96iT - 1.33e3T^{2} \)
17 \( 1 + 122.T + 4.91e3T^{2} \)
19 \( 1 + 43.1iT - 6.85e3T^{2} \)
23 \( 1 - 75.5T + 1.21e4T^{2} \)
29 \( 1 + 163.T + 2.43e4T^{2} \)
31 \( 1 + 139. iT - 2.97e4T^{2} \)
37 \( 1 - 2.80iT - 5.06e4T^{2} \)
41 \( 1 - 300. iT - 6.89e4T^{2} \)
43 \( 1 + 363.T + 7.95e4T^{2} \)
47 \( 1 + 41.2iT - 1.03e5T^{2} \)
53 \( 1 + 125.T + 1.48e5T^{2} \)
59 \( 1 + 407. iT - 2.05e5T^{2} \)
61 \( 1 - 536.T + 2.26e5T^{2} \)
67 \( 1 + 340. iT - 3.00e5T^{2} \)
71 \( 1 + 514. iT - 3.57e5T^{2} \)
73 \( 1 + 491. iT - 3.89e5T^{2} \)
79 \( 1 - 762.T + 4.93e5T^{2} \)
83 \( 1 + 345. iT - 5.71e5T^{2} \)
89 \( 1 - 362. iT - 7.04e5T^{2} \)
97 \( 1 - 276. iT - 9.12e5T^{2} \)
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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

−11.04547813344935467218698068863, −10.67885857550045043134723153760, −9.560852930478604839415351181212, −8.012661153220258731270791874567, −7.15844241816363592432373932917, −6.75934191482710474657462450333, −6.15891070322177281436783640939, −4.74643977173229549335308018838, −3.50940511709301479316688628251, −2.19293725959522632958770235059, 0.02860859180482005132891422410, 1.41978054510127668077786699809, 2.28635887410837357188529578987, 3.98203473018838202776665264399, 4.98903401593043494967622982456, 5.74286893230661031740468561525, 6.91484440481184037341462737951, 8.447449446166780313823681604518, 9.039429495705714709435517535783, 9.744552251917345304372208774927

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