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Properties

Label 2-507-13.4-c1-0-19
Degree $2$
Conductor $507$
Sign $-0.114 + 0.993i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.114 + 0.993i$
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: \((2,\ 507,\ (\ :1/2),\ -0.114 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.614791 - 0.689985i\)
\(L(\frac12)\) \(\approx\) \(0.614791 - 0.689985i\)
\(L(\frac{3}{2})\) 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

−10.71613045979604482107378119101, −10.04328282563969914532779597577, −8.876275667763187946156648086146, −8.133216584251195903834485900235, −6.77336010682308829244793312321, −5.82340390770154877502152166856, −4.93428156369790575453187745210, −4.25958159534674271864789747752, −2.63290388025957239082632648528, −0.51356859656640151804959839816, 1.98590194505190117031276684309, 3.16830665626593588014925429340, 4.80068316080582039943444404247, 5.11249379176899659657848214911, 6.69314405423215082351170247952, 7.62912959201150267339435052514, 8.370819162419067583300079149164, 9.141866934944685397948191189810, 10.59250197720214757031897280909, 11.15138373774175698715925143432

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