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Properties

Label 2-507-13.12-c3-0-27
Degree $2$
Conductor $507$
Sign $-0.277 - 0.960i$
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  + 4.12i·2-s − 3·3-s − 9·4-s − 3.05i·5-s − 12.3i·6-s + 6.68i·7-s − 4.12i·8-s + 9·9-s + 12.5·10-s − 32.2i·11-s + 27·12-s − 27.5·14-s + 9.15i·15-s − 55·16-s + 28.8·17-s + 37.1i·18-s + ⋯
L(s)  = 1  + 1.45i·2-s − 0.577·3-s − 1.12·4-s − 0.272i·5-s − 0.841i·6-s + 0.360i·7-s − 0.182i·8-s + 0.333·9-s + 0.397·10-s − 0.883i·11-s + 0.649·12-s − 0.526·14-s + 0.157i·15-s − 0.859·16-s + 0.411·17-s + 0.485i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.277 - 0.960i$
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: \((2,\ 507,\ (\ :3/2),\ -0.277 - 0.960i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.519494969\)
\(L(\frac12)\) \(\approx\) \(1.519494969\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.12iT - 8T^{2} \)
5 \( 1 + 3.05iT - 125T^{2} \)
7 \( 1 - 6.68iT - 343T^{2} \)
11 \( 1 + 32.2iT - 1.33e3T^{2} \)
17 \( 1 - 28.8T + 4.91e3T^{2} \)
19 \( 1 + 101. iT - 6.85e3T^{2} \)
23 \( 1 - 118.T + 1.21e4T^{2} \)
29 \( 1 - 160.T + 2.43e4T^{2} \)
31 \( 1 + 38.0iT - 2.97e4T^{2} \)
37 \( 1 - 327. iT - 5.06e4T^{2} \)
41 \( 1 - 56.0iT - 6.89e4T^{2} \)
43 \( 1 + 127.T + 7.95e4T^{2} \)
47 \( 1 - 517. iT - 1.03e5T^{2} \)
53 \( 1 + 695.T + 1.48e5T^{2} \)
59 \( 1 + 656. iT - 2.05e5T^{2} \)
61 \( 1 - 701.T + 2.26e5T^{2} \)
67 \( 1 + 57.1iT - 3.00e5T^{2} \)
71 \( 1 - 309. iT - 3.57e5T^{2} \)
73 \( 1 - 389. iT - 3.89e5T^{2} \)
79 \( 1 - 901.T + 4.93e5T^{2} \)
83 \( 1 - 687. iT - 5.71e5T^{2} \)
89 \( 1 + 1.07e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.75e3iT - 9.12e5T^{2} \)
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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

−10.90228126412125998073469600029, −9.544136386769674523357529948015, −8.702832904538426166680317201194, −8.005080914298824132518475147589, −6.84463824092277406030384069262, −6.31251880231694975893080498969, −5.24193588525661879466698730060, −4.71481401962390048473459176559, −2.92813762879801121316012322212, −0.828121668699752977646158261704, 0.77800311716528864538557325894, 1.91593430554675874171191260600, 3.20293690892617838063426145790, 4.20932840525056431537114674643, 5.21256761984273006692760158046, 6.60771376470093582973198682163, 7.41785088116631367900739766224, 8.808491673620562652901699306725, 9.849912040874679629049006654125, 10.42332569631937706071892921471

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