Properties

Label 2-507-1.1-c1-0-17
Degree $2$
Conductor $507$
Sign $1$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.35·2-s − 3-s + 3.55·4-s + 3.69·5-s − 2.35·6-s − 0.801·7-s + 3.66·8-s + 9-s + 8.70·10-s − 2.85·11-s − 3.55·12-s − 1.89·14-s − 3.69·15-s + 1.52·16-s + 2.93·17-s + 2.35·18-s − 2.44·19-s + 13.1·20-s + 0.801·21-s − 6.71·22-s − 7.78·23-s − 3.66·24-s + 8.63·25-s − 27-s − 2.85·28-s + 3.85·29-s − 8.70·30-s + ⋯
L(s)  = 1  + 1.66·2-s − 0.577·3-s + 1.77·4-s + 1.65·5-s − 0.962·6-s − 0.303·7-s + 1.29·8-s + 0.333·9-s + 2.75·10-s − 0.859·11-s − 1.02·12-s − 0.505·14-s − 0.953·15-s + 0.381·16-s + 0.712·17-s + 0.555·18-s − 0.560·19-s + 2.93·20-s + 0.174·21-s − 1.43·22-s − 1.62·23-s − 0.748·24-s + 1.72·25-s − 0.192·27-s − 0.538·28-s + 0.715·29-s − 1.58·30-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(3.515463422\)
\(L(\frac12)\) \(\approx\) \(3.515463422\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
13 \( 1 \)
good2 \( 1 - 2.35T + 2T^{2} \)
5 \( 1 - 3.69T + 5T^{2} \)
7 \( 1 + 0.801T + 7T^{2} \)
11 \( 1 + 2.85T + 11T^{2} \)
17 \( 1 - 2.93T + 17T^{2} \)
19 \( 1 + 2.44T + 19T^{2} \)
23 \( 1 + 7.78T + 23T^{2} \)
29 \( 1 - 3.85T + 29T^{2} \)
31 \( 1 - 2.34T + 31T^{2} \)
37 \( 1 - 7.44T + 37T^{2} \)
41 \( 1 + 0.850T + 41T^{2} \)
43 \( 1 + 1.61T + 43T^{2} \)
47 \( 1 - 2.44T + 47T^{2} \)
53 \( 1 + 9.96T + 53T^{2} \)
59 \( 1 + 5.38T + 59T^{2} \)
61 \( 1 + 13.2T + 61T^{2} \)
67 \( 1 - 14.3T + 67T^{2} \)
71 \( 1 - 8.12T + 71T^{2} \)
73 \( 1 + 11.8T + 73T^{2} \)
79 \( 1 - 5.40T + 79T^{2} \)
83 \( 1 - 7.04T + 83T^{2} \)
89 \( 1 - 1.13T + 89T^{2} \)
97 \( 1 + 5.94T + 97T^{2} \)
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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

−11.02428000056674997789312650067, −10.22851001352422597459392482860, −9.541523968179841066512171474452, −7.931350372598345093761783796969, −6.49836467154880290748927830507, −6.08007829497662988317020565119, −5.34355285688175209912160995369, −4.47555183432868305616718971206, −3.00726401005179861311751062407, −1.96556914197670966460621889314, 1.96556914197670966460621889314, 3.00726401005179861311751062407, 4.47555183432868305616718971206, 5.34355285688175209912160995369, 6.08007829497662988317020565119, 6.49836467154880290748927830507, 7.931350372598345093761783796969, 9.541523968179841066512171474452, 10.22851001352422597459392482860, 11.02428000056674997789312650067

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