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Properties

Label 2-507-507.371-c1-0-10
Degree $2$
Conductor $507$
Sign $0.496 - 0.868i$
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  + (−0.346 − 1.69i)3-s + (−1.92 + 0.556i)4-s + (−0.766 + 1.06i)7-s + (−2.75 + 1.17i)9-s + (1.60 + 3.06i)12-s + (2.59 + 2.5i)13-s + (3.38 − 2.13i)16-s + (−2.18 + 8.14i)19-s + (2.07 + 0.932i)21-s + (4.11 + 2.84i)25-s + (2.95 + 4.27i)27-s + (0.880 − 2.47i)28-s + (0.683 − 3.72i)31-s + (4.64 − 3.79i)36-s + (−1.43 + 1.68i)37-s + ⋯
L(s)  = 1  + (−0.200 − 0.979i)3-s + (−0.960 + 0.278i)4-s + (−0.289 + 0.402i)7-s + (−0.919 + 0.391i)9-s + (0.464 + 0.885i)12-s + (0.720 + 0.693i)13-s + (0.845 − 0.534i)16-s + (−0.500 + 1.86i)19-s + (0.452 + 0.203i)21-s + (0.822 + 0.568i)25-s + (0.568 + 0.822i)27-s + (0.166 − 0.467i)28-s + (0.122 − 0.669i)31-s + (0.774 − 0.632i)36-s + (−0.235 + 0.277i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 - 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.496 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.496 - 0.868i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.496 - 0.868i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.647625 + 0.375804i\)
\(L(\frac12)\) \(\approx\) \(0.647625 + 0.375804i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.346 + 1.69i)T \)
13 \( 1 + (-2.59 - 2.5i)T \)
good2 \( 1 + (1.92 - 0.556i)T^{2} \)
5 \( 1 + (-4.11 - 2.84i)T^{2} \)
7 \( 1 + (0.766 - 1.06i)T + (-2.21 - 6.63i)T^{2} \)
11 \( 1 + (-7.93 + 7.61i)T^{2} \)
17 \( 1 + (16.1 - 5.38i)T^{2} \)
19 \( 1 + (2.18 - 8.14i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-8.06 - 27.8i)T^{2} \)
31 \( 1 + (-0.683 + 3.72i)T + (-28.9 - 10.9i)T^{2} \)
37 \( 1 + (1.43 - 1.68i)T + (-5.93 - 36.5i)T^{2} \)
41 \( 1 + (16.0 + 37.7i)T^{2} \)
43 \( 1 + (0.928 - 11.5i)T + (-42.4 - 6.89i)T^{2} \)
47 \( 1 + (21.8 + 41.6i)T^{2} \)
53 \( 1 + (39.6 + 35.1i)T^{2} \)
59 \( 1 + (-53.3 + 25.2i)T^{2} \)
61 \( 1 + (-3.35 + 4.10i)T + (-12.2 - 59.7i)T^{2} \)
67 \( 1 + (7.56 - 4.16i)T + (35.8 - 56.6i)T^{2} \)
71 \( 1 + (42.6 - 56.7i)T^{2} \)
73 \( 1 + (12.3 + 9.69i)T + (17.4 + 70.8i)T^{2} \)
79 \( 1 + (17.1 - 4.23i)T + (69.9 - 36.7i)T^{2} \)
83 \( 1 + (82.3 - 10.0i)T^{2} \)
89 \( 1 + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (6.73 - 7.30i)T + (-7.80 - 96.6i)T^{2} \)
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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

−11.24887138980594540720290734172, −10.09833152768191883044348392592, −9.056774664268521126628137788369, −8.379100880349567959697834850792, −7.57846225759762665634277481440, −6.35034326071257822508051028506, −5.65352376523218110671037296416, −4.34569472074884236396898836272, −3.12411760467240664614022308271, −1.47040979894968778713442361123, 0.50614232747603965924321159786, 3.06568575638942769060630388121, 4.13228477896875046447316558231, 4.94543704230641976141737571967, 5.85153282994503330473035488761, 7.01433205575507168885612296237, 8.721071649013090104968440371462, 8.760275297394681363935122751267, 10.06688843959752743131478207323, 10.50097391033457690359391720427

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