Properties

Label 2-76-4.3-c2-0-16
Degree $2$
Conductor $76$
Sign $-0.665 + 0.746i$
Analytic cond. $2.07085$
Root an. cond. $1.43904$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.711 − 1.86i)2-s − 4.44i·3-s + (−2.98 − 2.66i)4-s + 4.97·5-s + (−8.31 − 3.16i)6-s + 12.2i·7-s + (−7.09 + 3.68i)8-s − 10.7·9-s + (3.54 − 9.30i)10-s − 13.4i·11-s + (−11.8 + 13.2i)12-s + 14.1·13-s + (22.9 + 8.72i)14-s − 22.1i·15-s + (1.84 + 15.8i)16-s − 5.89·17-s + ⋯
L(s)  = 1  + (0.355 − 0.934i)2-s − 1.48i·3-s + (−0.746 − 0.665i)4-s + 0.995·5-s + (−1.38 − 0.527i)6-s + 1.75i·7-s + (−0.887 + 0.461i)8-s − 1.19·9-s + (0.354 − 0.930i)10-s − 1.22i·11-s + (−0.986 + 1.10i)12-s + 1.09·13-s + (1.63 + 0.623i)14-s − 1.47i·15-s + (0.115 + 0.993i)16-s − 0.346·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.665 + 0.746i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.665 + 0.746i$
Analytic conductor: \(2.07085\)
Root analytic conductor: \(1.43904\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1),\ -0.665 + 0.746i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.629294 - 1.40331i\)
\(L(\frac12)\) \(\approx\) \(0.629294 - 1.40331i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.711 + 1.86i)T \)
19 \( 1 - 4.35iT \)
good3 \( 1 + 4.44iT - 9T^{2} \)
5 \( 1 - 4.97T + 25T^{2} \)
7 \( 1 - 12.2iT - 49T^{2} \)
11 \( 1 + 13.4iT - 121T^{2} \)
13 \( 1 - 14.1T + 169T^{2} \)
17 \( 1 + 5.89T + 289T^{2} \)
23 \( 1 - 0.906iT - 529T^{2} \)
29 \( 1 + 10.3T + 841T^{2} \)
31 \( 1 - 43.2iT - 961T^{2} \)
37 \( 1 + 1.61T + 1.36e3T^{2} \)
41 \( 1 - 69.3T + 1.68e3T^{2} \)
43 \( 1 + 32.0iT - 1.84e3T^{2} \)
47 \( 1 - 38.9iT - 2.20e3T^{2} \)
53 \( 1 + 8.31T + 2.80e3T^{2} \)
59 \( 1 - 20.9iT - 3.48e3T^{2} \)
61 \( 1 + 118.T + 3.72e3T^{2} \)
67 \( 1 - 57.4iT - 4.48e3T^{2} \)
71 \( 1 - 11.3iT - 5.04e3T^{2} \)
73 \( 1 + 23.5T + 5.32e3T^{2} \)
79 \( 1 + 0.286iT - 6.24e3T^{2} \)
83 \( 1 + 24.9iT - 6.88e3T^{2} \)
89 \( 1 + 43.7T + 7.92e3T^{2} \)
97 \( 1 - 115.T + 9.40e3T^{2} \)
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

−13.56309194621197621046214009712, −12.78780755213878974139721285513, −11.93848127309956208454493169848, −10.92497541056863234796971560281, −9.171365861564149332752039070045, −8.450622197990576321043542634664, −6.12566859038458581874836167892, −5.72808470595039932793024412330, −2.81400489300831810842078926254, −1.58423177261008336429867314971, 3.87307261716257803183768788115, 4.68022535086379776671605372582, 6.18029446098285927461373153494, 7.56178870720052699032033913751, 9.270943798168117405637474343137, 9.963148469414577129575517954905, 10.98142586055868705943982180222, 13.06771283300753005460493259802, 13.79963319909263615918754661917, 14.74822181719587093372212541455

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