Properties

Degree 2
Conductor 751
Sign $1$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.41i·3-s − 1.41i·6-s − 1.41i·7-s + 8-s − 1.00·9-s − 1.41i·11-s + 13-s + 1.41i·14-s − 16-s − 1.41i·17-s + 1.00·18-s + 19-s + 2.00·21-s + 1.41i·22-s − 23-s + ⋯
L(s)  = 1  − 2-s + 1.41i·3-s − 1.41i·6-s − 1.41i·7-s + 8-s − 1.00·9-s − 1.41i·11-s + 13-s + 1.41i·14-s − 16-s − 1.41i·17-s + 1.00·18-s + 19-s + 2.00·21-s + 1.41i·22-s − 23-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(751\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{751} (750, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 751,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.5381923100$
$L(\frac12)$  $\approx$  $0.5381923100$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 751$, \(F_p\) is a polynomial of degree 2. If $p = 751$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad751 \( 1 - T \)
good2 \( 1 + T + T^{2} \)
3 \( 1 - 1.41iT - T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + 1.41iT - T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + T + T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + 1.41iT - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 - T + T^{2} \)
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

−10.35651866957979720096358036271, −9.673928885295727625658486278934, −9.171626692946047428752880614050, −8.128082750215580302253367260231, −7.51355238980620300435290255142, −6.13439928409172316706261872110, −4.94707303497722757937535216993, −4.05723600085610586457565379139, −3.33661979652402563438587541061, −0.914073185358732862521733003437, 1.55412332810692622166085780454, 2.20845489641497744547796932775, 4.08468593395387771530151555318, 5.58255149040677268423848842995, 6.32261504508002799283594173611, 7.43538701570160883080469817692, 7.996058707554852262575217903770, 8.719758964126735204866411302666, 9.517163490604226734078892592058, 10.34261454167724309566002468702

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