Properties

Label 2-507-13.12-c1-0-2
Degree $2$
Conductor $507$
Sign $0.691 + 0.722i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.69i·2-s − 3-s − 5.24·4-s + 1.04i·5-s + 2.69i·6-s + 0.554i·7-s + 8.74i·8-s + 9-s + 2.82·10-s + 2.91i·11-s + 5.24·12-s + 1.49·14-s − 1.04i·15-s + 13.0·16-s + 4.85·17-s − 2.69i·18-s + ⋯
L(s)  = 1  − 1.90i·2-s − 0.577·3-s − 2.62·4-s + 0.469i·5-s + 1.09i·6-s + 0.209i·7-s + 3.09i·8-s + 0.333·9-s + 0.892·10-s + 0.877i·11-s + 1.51·12-s + 0.399·14-s − 0.270i·15-s + 3.25·16-s + 1.17·17-s − 0.634i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.691 + 0.722i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.691 + 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.762762 - 0.325622i\)
\(L(\frac12)\) \(\approx\) \(0.762762 - 0.325622i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.69iT - 2T^{2} \)
5 \( 1 - 1.04iT - 5T^{2} \)
7 \( 1 - 0.554iT - 7T^{2} \)
11 \( 1 - 2.91iT - 11T^{2} \)
17 \( 1 - 4.85T + 17T^{2} \)
19 \( 1 - 0.753iT - 19T^{2} \)
23 \( 1 + 5.76T + 23T^{2} \)
29 \( 1 + 1.91T + 29T^{2} \)
31 \( 1 - 9.51iT - 31T^{2} \)
37 \( 1 - 5.75iT - 37T^{2} \)
41 \( 1 + 4.91iT - 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 - 0.753iT - 47T^{2} \)
53 \( 1 + 7.58T + 53T^{2} \)
59 \( 1 - 4.09iT - 59T^{2} \)
61 \( 1 + 3.42T + 61T^{2} \)
67 \( 1 - 1.87iT - 67T^{2} \)
71 \( 1 - 10.5iT - 71T^{2} \)
73 \( 1 - 10.4iT - 73T^{2} \)
79 \( 1 - 1.33T + 79T^{2} \)
83 \( 1 + 2.64iT - 83T^{2} \)
89 \( 1 - 9.92iT - 89T^{2} \)
97 \( 1 + 17.0iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.73873219059207446410156089213, −10.22733530994332378678351123597, −9.538461493695778912183071773518, −8.476858772976323675348507410648, −7.27458693931038230018875232387, −5.76165902909497038600322071581, −4.77739702292958854066591356164, −3.74934270031281995409354816353, −2.62928793339349890588837605010, −1.35479947325132124480639002243, 0.61856969494096997661188266923, 3.78191431444966897064624159984, 4.76981986409579124471295970560, 5.78404864299300655338576803600, 6.17014533970443346282452374444, 7.47890302062870741140047292776, 7.968082557174193774427173795837, 9.023076941521660372795479791779, 9.745704545108987706096917789871, 10.87704805588224926532475307568

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