Properties

Label 2-507-39.5-c1-0-7
Degree $2$
Conductor $507$
Sign $0.0450 + 0.998i$
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 + (1.19 + 3.28i)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 + (−1.44 − 3.96i)15-s + 3.84·16-s + 5.78·17-s + (0.515 − 6.03i)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 + (0.489 + 1.34i)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 + (−0.373 − 1.02i)15-s + 0.961·16-s + 1.40·17-s + (0.121 − 1.42i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0450 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0450 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.0450 + 0.998i$
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: \((2,\ 507,\ (\ :1/2),\ 0.0450 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.438568 - 0.419226i\)
\(L(\frac12)\) \(\approx\) \(0.438568 - 0.419226i\)
\(L(\frac{3}{2})\) 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

−10.54663222580306048055134419710, −10.02801096958636131013272000831, −9.521487542188233920553263081802, −7.938348927425553897573909593706, −7.12361287653074844043074653639, −6.23244671441294247383288114387, −5.16168448942085024887371820339, −3.35971577100309043524689253930, −2.21727577023369613238101656973, −0.808250397132473998634649886204, 0.937701419882341279135477638063, 3.24680232074705900433647398860, 5.14089930833169014017195613128, 5.75477434223663301283514047509, 6.34099837778122629448953090478, 7.52675835069519305041710103954, 8.621992467229916376774511515472, 9.377451333626533810583329227934, 9.855805476249862688426890413802, 10.73228194014305922544313535526

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