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Properties

Label 2-507-507.53-c0-0-0
Degree $2$
Conductor $507$
Sign $-0.389 + 0.921i$
Analytic cond. $0.253025$
Root an. cond. $0.503016$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.354 − 0.935i)3-s + (0.120 − 0.992i)4-s + (−0.402 − 0.583i)7-s + (−0.748 + 0.663i)9-s + (−0.970 + 0.239i)12-s + 13-s + (−0.970 − 0.239i)16-s − 1.49·19-s + (−0.402 + 0.583i)21-s + (0.885 − 0.464i)25-s + (0.885 + 0.464i)27-s + (−0.627 + 0.329i)28-s + (0.213 + 0.112i)31-s + (0.568 + 0.822i)36-s + (1.00 + 0.527i)37-s + ⋯
L(s)  = 1  + (−0.354 − 0.935i)3-s + (0.120 − 0.992i)4-s + (−0.402 − 0.583i)7-s + (−0.748 + 0.663i)9-s + (−0.970 + 0.239i)12-s + 13-s + (−0.970 − 0.239i)16-s − 1.49·19-s + (−0.402 + 0.583i)21-s + (0.885 − 0.464i)25-s + (0.885 + 0.464i)27-s + (−0.627 + 0.329i)28-s + (0.213 + 0.112i)31-s + (0.568 + 0.822i)36-s + (1.00 + 0.527i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.389 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.389 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.389 + 0.921i$
Analytic conductor: \(0.253025\)
Root analytic conductor: \(0.503016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :0),\ -0.389 + 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7460663999\)
\(L(\frac12)\) \(\approx\) \(0.7460663999\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.354 + 0.935i)T \)
13 \( 1 - T \)
good2 \( 1 + (-0.120 + 0.992i)T^{2} \)
5 \( 1 + (-0.885 + 0.464i)T^{2} \)
7 \( 1 + (0.402 + 0.583i)T + (-0.354 + 0.935i)T^{2} \)
11 \( 1 + (-0.120 - 0.992i)T^{2} \)
17 \( 1 + (0.354 - 0.935i)T^{2} \)
19 \( 1 + 1.49T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.120 + 0.992i)T^{2} \)
31 \( 1 + (-0.213 - 0.112i)T + (0.568 + 0.822i)T^{2} \)
37 \( 1 + (-1.00 - 0.527i)T + (0.568 + 0.822i)T^{2} \)
41 \( 1 + (0.748 - 0.663i)T^{2} \)
43 \( 1 + (-1.00 + 0.527i)T + (0.568 - 0.822i)T^{2} \)
47 \( 1 + (0.970 - 0.239i)T^{2} \)
53 \( 1 + (0.354 - 0.935i)T^{2} \)
59 \( 1 + (-0.885 + 0.464i)T^{2} \)
61 \( 1 + (-0.136 + 0.198i)T + (-0.354 - 0.935i)T^{2} \)
67 \( 1 + (-0.213 - 1.75i)T + (-0.970 + 0.239i)T^{2} \)
71 \( 1 + (0.748 - 0.663i)T^{2} \)
73 \( 1 + (-1.45 - 1.28i)T + (0.120 + 0.992i)T^{2} \)
79 \( 1 + (0.0854 + 0.704i)T + (-0.970 + 0.239i)T^{2} \)
83 \( 1 + (0.748 + 0.663i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (1.94 + 0.478i)T + (0.885 + 0.464i)T^{2} \)
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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

−10.80497368920817241065872828942, −10.27826684636277369086134527853, −9.004352874808689198342372694956, −8.129789414451513186028515266074, −6.82743187176155062373092549342, −6.41919199410719905138696517062, −5.47460640665922595325836933171, −4.20301604490796133262241565994, −2.45180197848209819701051009766, −1.04282946525708406825762522373, 2.64773692660318462040311195524, 3.69741831928821093657626268223, 4.56785070019272463428865655729, 5.91370060988832605402936132986, 6.64414431714625215471032294613, 8.056528852325766186550253666151, 8.839277528897152977188612202094, 9.461281381815656167705409806376, 10.82406047696312149161294787593, 11.14971544714449069709582568287

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