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Properties

Label 2-507-39.5-c1-0-3
Degree $2$
Conductor $507$
Sign $0.0145 - 0.999i$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
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  + (−1.42 − 1.42i)2-s + (−1.57 + 0.730i)3-s + 2.07i·4-s + (1.72 + 1.72i)5-s + (3.28 + 1.19i)6-s + (2.20 + 2.20i)7-s + (0.105 − 0.105i)8-s + (1.93 − 2.29i)9-s − 4.91i·10-s + (−1.95 + 1.95i)11-s + (−1.51 − 3.25i)12-s − 6.28i·14-s + (−3.96 − 1.44i)15-s + 3.84·16-s − 5.78·17-s + (−6.03 + 0.515i)18-s + ⋯
L(s)  = 1  + (−1.00 − 1.00i)2-s + (−0.906 + 0.421i)3-s + 1.03i·4-s + (0.770 + 0.770i)5-s + (1.34 + 0.489i)6-s + (0.831 + 0.831i)7-s + (0.0372 − 0.0372i)8-s + (0.644 − 0.764i)9-s − 1.55i·10-s + (−0.590 + 0.590i)11-s + (−0.437 − 0.940i)12-s − 1.67i·14-s + (−1.02 − 0.373i)15-s + 0.961·16-s − 1.40·17-s + (−1.42 + 0.121i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0145 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0145 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.0145 - 0.999i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.0145 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.328734 + 0.323969i\)
\(L(\frac12)\) \(\approx\) \(0.328734 + 0.323969i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.57 - 0.730i)T \)
13 \( 1 \)
good2 \( 1 + (1.42 + 1.42i)T + 2iT^{2} \)
5 \( 1 + (-1.72 - 1.72i)T + 5iT^{2} \)
7 \( 1 + (-2.20 - 2.20i)T + 7iT^{2} \)
11 \( 1 + (1.95 - 1.95i)T - 11iT^{2} \)
17 \( 1 + 5.78T + 17T^{2} \)
19 \( 1 + (1.06 - 1.06i)T - 19iT^{2} \)
23 \( 1 + 3.86T + 23T^{2} \)
29 \( 1 + 2.92iT - 29T^{2} \)
31 \( 1 + (3.56 - 3.56i)T - 31iT^{2} \)
37 \( 1 + (-2.83 - 2.83i)T + 37iT^{2} \)
41 \( 1 + (-4.79 - 4.79i)T + 41iT^{2} \)
43 \( 1 + 1.84iT - 43T^{2} \)
47 \( 1 + (-0.115 + 0.115i)T - 47iT^{2} \)
53 \( 1 - 10.0iT - 53T^{2} \)
59 \( 1 + (1.22 - 1.22i)T - 59iT^{2} \)
61 \( 1 + 5.39T + 61T^{2} \)
67 \( 1 + (9.65 - 9.65i)T - 67iT^{2} \)
71 \( 1 + (0.239 + 0.239i)T + 71iT^{2} \)
73 \( 1 + (8.54 + 8.54i)T + 73iT^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 + (-2.83 - 2.83i)T + 83iT^{2} \)
89 \( 1 + (-3.00 + 3.00i)T - 89iT^{2} \)
97 \( 1 + (6.99 - 6.99i)T - 97iT^{2} \)
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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

−10.80611093920414794995824449297, −10.48891496299737237913382887208, −9.598490055641155631611235548226, −8.884257893180446111841019160259, −7.75062963446785471150068928539, −6.40557452476354963541770319709, −5.59502183345908268390448849414, −4.43336770451677560977975615028, −2.64438751125992213647916930452, −1.79113811431017352252476764307, 0.42073744241789426135081391117, 1.78067541160611119220414621616, 4.39779052179380648687511603467, 5.42845400978754467046254237867, 6.17157994583765956409065302635, 7.15531129566005872930949563341, 7.899941935700200335353720479503, 8.733579313362687662688405364431, 9.636408120232575363680695167022, 10.67069578915175867253752703762

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