Properties

Label 2-507-13.4-c1-0-8
Degree $2$
Conductor $507$
Sign $0.944 + 0.327i$
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  + (−0.385 − 0.222i)2-s + (−0.5 + 0.866i)3-s + (−0.900 − 1.56i)4-s − 0.246i·5-s + (0.385 − 0.222i)6-s + (−1.51 + 0.876i)7-s + 1.69i·8-s + (−0.499 − 0.866i)9-s + (−0.0549 + 0.0951i)10-s + (4.89 + 2.82i)11-s + 1.80·12-s + 0.780·14-s + (0.213 + 0.123i)15-s + (−1.42 + 2.46i)16-s + (−1.90 − 3.29i)17-s + 0.445i·18-s + ⋯
L(s)  = 1  + (−0.272 − 0.157i)2-s + (−0.288 + 0.499i)3-s + (−0.450 − 0.780i)4-s − 0.110i·5-s + (0.157 − 0.0908i)6-s + (−0.573 + 0.331i)7-s + 0.598i·8-s + (−0.166 − 0.288i)9-s + (−0.0173 + 0.0301i)10-s + (1.47 + 0.852i)11-s + 0.520·12-s + 0.208·14-s + (0.0552 + 0.0318i)15-s + (−0.356 + 0.617i)16-s + (−0.461 − 0.798i)17-s + 0.104i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.944 + 0.327i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.944 + 0.327i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.993033 - 0.167439i\)
\(L(\frac12)\) \(\approx\) \(0.993033 - 0.167439i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (0.385 + 0.222i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 0.246iT - 5T^{2} \)
7 \( 1 + (1.51 - 0.876i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.89 - 2.82i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.90 + 3.29i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.83 + 2.79i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.17 + 7.22i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.96 + 5.14i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.26iT - 31T^{2} \)
37 \( 1 + (-2.76 - 1.59i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.385 + 0.222i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.856 - 1.48i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.73iT - 47T^{2} \)
53 \( 1 + 1.06T + 53T^{2} \)
59 \( 1 + (-11.8 + 6.85i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.25 - 7.37i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.16 + 2.98i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.95 - 2.85i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.35iT - 73T^{2} \)
79 \( 1 - 4.45T + 79T^{2} \)
83 \( 1 + 10.1iT - 83T^{2} \)
89 \( 1 + (-0.118 - 0.0685i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (11.8 - 6.84i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.76264392909579892476641162456, −9.770488072029371480900139810156, −9.300859192099910910249205791461, −8.669090456340692870676537400917, −6.94499202666706841810895306522, −6.29389977403937742273853032692, −4.99332609922095307352108622564, −4.41541194307444616245635208037, −2.78245625435021061829802838446, −0.960492078174909365181795592660, 1.08942768258099858910050036241, 3.23334605440969161790327674890, 3.95282526567200115913145536904, 5.49622760062448884555473659400, 6.64956668940361359272330206936, 7.20154243022886142076631034587, 8.322312698988666460207554545180, 9.062790619244924190852878832946, 9.879804735010014335325649209806, 11.15721665176891362752950688407

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