Properties

Degree 2
Conductor $ 2^{3} \cdot 7^{2} $
Sign $-0.850 - 0.526i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.104 − 1.99i)2-s + (−1.99 + 3.44i)3-s + (−3.97 − 0.418i)4-s + (1.63 − 0.941i)5-s + (6.67 + 4.33i)6-s + (−1.25 + 7.90i)8-s + (−3.42 − 5.93i)9-s + (−1.70 − 3.35i)10-s + (3.93 − 6.82i)11-s + (9.36 − 12.8i)12-s + 11.4i·13-s + 7.49i·15-s + (15.6 + 3.33i)16-s + (−1.44 + 2.51i)17-s + (−12.2 + 6.21i)18-s + (−15.0 − 26.0i)19-s + ⋯
L(s)  = 1  + (0.0523 − 0.998i)2-s + (−0.663 + 1.14i)3-s + (−0.994 − 0.104i)4-s + (0.326 − 0.188i)5-s + (1.11 + 0.722i)6-s + (−0.156 + 0.987i)8-s + (−0.380 − 0.659i)9-s + (−0.170 − 0.335i)10-s + (0.358 − 0.620i)11-s + (0.780 − 1.07i)12-s + 0.883i·13-s + 0.499i·15-s + (0.978 + 0.208i)16-s + (−0.0852 + 0.147i)17-s + (−0.678 + 0.345i)18-s + (−0.790 − 1.36i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(392\)    =    \(2^{3} \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.850 - 0.526i$
motivic weight  =  \(2\)
character  :  $\chi_{392} (67, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 392,\ (\ :1),\ -0.850 - 0.526i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0432959 + 0.152250i\)
\(L(\frac12)\)  \(\approx\)  \(0.0432959 + 0.152250i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-0.104 + 1.99i)T \)
7 \( 1 \)
good3 \( 1 + (1.99 - 3.44i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (-1.63 + 0.941i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-3.93 + 6.82i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 11.4iT - 169T^{2} \)
17 \( 1 + (1.44 - 2.51i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (15.0 + 26.0i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (33.3 - 19.2i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 27.8iT - 841T^{2} \)
31 \( 1 + (19.4 + 11.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (39.4 - 22.7i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 40.6T + 1.68e3T^{2} \)
43 \( 1 + 47.2T + 1.84e3T^{2} \)
47 \( 1 + (71.5 - 41.2i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (23.2 + 13.4i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (5.20 - 9.01i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-19.1 + 11.0i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-29.6 + 51.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 38.2iT - 5.04e3T^{2} \)
73 \( 1 + (-6.98 + 12.1i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-44.3 + 25.5i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 89.4T + 6.88e3T^{2} \)
89 \( 1 + (-52.6 - 91.1i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 55.3T + 9.40e3T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.33591594173680574529167751219, −10.72795273891137294140157467659, −9.604888423454534006213913452927, −9.352478014002386995435134694421, −8.145854469675208165051929847418, −6.33879196311532333328021445941, −5.33663599818435858334653196853, −4.42382875299850012517327037750, −3.60200047011411358575192801801, −1.90670107170903811493783003213, 0.07279213038137324653108910375, 1.78987662684924012620950189231, 3.85806735082988027533117758284, 5.27869336034853662789172387477, 6.17452464957165692493740749868, 6.73594200411639668474330129845, 7.73283139664120342672402743128, 8.449971709080178797067560564157, 9.809065245750300892912816446397, 10.56483273485627291523038110273

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