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Properties

Label 2-507-39.8-c1-0-35
Degree $2$
Conductor $507$
Sign $0.0809 + 0.996i$
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  + (1.69 − 1.69i)2-s + (−0.366 + 1.69i)3-s − 3.73i·4-s + (1.69 − 1.69i)5-s + (2.24 + 3.48i)6-s + (1 − i)7-s + (−2.93 − 2.93i)8-s + (−2.73 − 1.23i)9-s − 5.73i·10-s + (1.23 + 1.23i)11-s + (6.31 + 1.36i)12-s − 3.38i·14-s + (2.24 + 3.48i)15-s − 2.46·16-s − 2.14·17-s + (−6.72 + 2.52i)18-s + ⋯
L(s)  = 1  + (1.19 − 1.19i)2-s + (−0.211 + 0.977i)3-s − 1.86i·4-s + (0.757 − 0.757i)5-s + (0.917 + 1.42i)6-s + (0.377 − 0.377i)7-s + (−1.03 − 1.03i)8-s + (−0.910 − 0.413i)9-s − 1.81i·10-s + (0.373 + 0.373i)11-s + (1.82 + 0.394i)12-s − 0.904i·14-s + (0.580 + 0.899i)15-s − 0.616·16-s − 0.520·17-s + (−1.58 + 0.595i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0809 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0809 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.0809 + 0.996i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.0809 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04281 - 1.88357i\)
\(L(\frac12)\) \(\approx\) \(2.04281 - 1.88357i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.366 - 1.69i)T \)
13 \( 1 \)
good2 \( 1 + (-1.69 + 1.69i)T - 2iT^{2} \)
5 \( 1 + (-1.69 + 1.69i)T - 5iT^{2} \)
7 \( 1 + (-1 + i)T - 7iT^{2} \)
11 \( 1 + (-1.23 - 1.23i)T + 11iT^{2} \)
17 \( 1 + 2.14T + 17T^{2} \)
19 \( 1 + (0.732 + 0.732i)T + 19iT^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 5.53iT - 29T^{2} \)
31 \( 1 + (-4.46 - 4.46i)T + 31iT^{2} \)
37 \( 1 + (4.83 - 4.83i)T - 37iT^{2} \)
41 \( 1 + (0.453 - 0.453i)T - 41iT^{2} \)
43 \( 1 - 8.19iT - 43T^{2} \)
47 \( 1 + (6.77 + 6.77i)T + 47iT^{2} \)
53 \( 1 - 4.62iT - 53T^{2} \)
59 \( 1 + (-3.38 - 3.38i)T + 59iT^{2} \)
61 \( 1 + 7T + 61T^{2} \)
67 \( 1 + (-6.19 - 6.19i)T + 67iT^{2} \)
71 \( 1 + (3.38 - 3.38i)T - 71iT^{2} \)
73 \( 1 + (6.09 - 6.09i)T - 73iT^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + (1.23 - 1.23i)T - 83iT^{2} \)
89 \( 1 + (-7.10 - 7.10i)T + 89iT^{2} \)
97 \( 1 + (-9.19 - 9.19i)T + 97iT^{2} \)
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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

−10.82923638207396445857889857738, −10.04127506622874608156560958333, −9.431138584018305743329955415144, −8.368381556001954407189146315008, −6.48601875866389231109836527807, −5.46178848330872228734119450473, −4.71609598732893158752847633732, −4.11078235732421747882092760109, −2.81146271871794290744135972907, −1.42527144426796813949704219879, 2.10265355031866332586722121472, 3.36559672767332452165088586022, 4.88338450305186718858226156639, 5.80715997734527567379199314418, 6.41952744305079114332801955552, 7.06267455031392739395373305675, 8.034977275689228911857854014208, 8.913647372841270370225569311534, 10.44912926369867296962887075926, 11.45872541136479552848032836978

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