Properties

Degree $2$
Conductor $8034$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 0.643·5-s + 6-s + 1.17·7-s − 8-s + 9-s − 0.643·10-s − 1.62·11-s − 12-s − 13-s − 1.17·14-s − 0.643·15-s + 16-s + 1.49·17-s − 18-s + 6.90·19-s + 0.643·20-s − 1.17·21-s + 1.62·22-s − 6.05·23-s + 24-s − 4.58·25-s + 26-s − 27-s + 1.17·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.287·5-s + 0.408·6-s + 0.444·7-s − 0.353·8-s + 0.333·9-s − 0.203·10-s − 0.489·11-s − 0.288·12-s − 0.277·13-s − 0.314·14-s − 0.166·15-s + 0.250·16-s + 0.363·17-s − 0.235·18-s + 1.58·19-s + 0.143·20-s − 0.256·21-s + 0.345·22-s − 1.26·23-s + 0.204·24-s − 0.917·25-s + 0.196·26-s − 0.192·27-s + 0.222·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8034\)    =    \(2 \cdot 3 \cdot 13 \cdot 103\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{8034} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8034,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.173959610\)
\(L(\frac12)\) \(\approx\) \(1.173959610\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
13 \( 1 + T \)
103 \( 1 - T \)
good5 \( 1 - 0.643T + 5T^{2} \)
7 \( 1 - 1.17T + 7T^{2} \)
11 \( 1 + 1.62T + 11T^{2} \)
17 \( 1 - 1.49T + 17T^{2} \)
19 \( 1 - 6.90T + 19T^{2} \)
23 \( 1 + 6.05T + 23T^{2} \)
29 \( 1 - 6.15T + 29T^{2} \)
31 \( 1 + 2.72T + 31T^{2} \)
37 \( 1 - 9.41T + 37T^{2} \)
41 \( 1 + 7.56T + 41T^{2} \)
43 \( 1 - 8.05T + 43T^{2} \)
47 \( 1 + 6.76T + 47T^{2} \)
53 \( 1 + 5.26T + 53T^{2} \)
59 \( 1 - 8.48T + 59T^{2} \)
61 \( 1 - 12.9T + 61T^{2} \)
67 \( 1 + 3.48T + 67T^{2} \)
71 \( 1 - 9.90T + 71T^{2} \)
73 \( 1 - 2.07T + 73T^{2} \)
79 \( 1 + 0.329T + 79T^{2} \)
83 \( 1 + 1.69T + 83T^{2} \)
89 \( 1 + 1.76T + 89T^{2} \)
97 \( 1 + 15.6T + 97T^{2} \)
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

−7.997838727422297793421722700948, −7.23744516661871728468297405371, −6.50433772993526914612743809236, −5.71645669138913484029058930453, −5.27402170998260257380827122263, −4.39663715246032808251734822549, −3.40627668313971959404119847739, −2.45150436432509136692475059060, −1.59768554207614817148655157060, −0.63282210881924992784611707381, 0.63282210881924992784611707381, 1.59768554207614817148655157060, 2.45150436432509136692475059060, 3.40627668313971959404119847739, 4.39663715246032808251734822549, 5.27402170998260257380827122263, 5.71645669138913484029058930453, 6.50433772993526914612743809236, 7.23744516661871728468297405371, 7.997838727422297793421722700948

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