↑

Properties

Label 2-756-63.5-c1-0-4
Degree $2$
Conductor $756$
Sign $0.749 - 0.662i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
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  + (1.95 + 3.39i)5-s + (0.554 − 2.58i)7-s + (3.19 + 1.84i)11-s + (0.480 + 0.277i)13-s + (−2.91 − 5.05i)17-s + (4.62 + 2.66i)19-s + (1.96 − 1.13i)23-s + (−5.16 + 8.94i)25-s + (−3.53 + 2.04i)29-s + 8.08i·31-s + (9.85 − 3.18i)35-s + (3.89 − 6.75i)37-s + (−3.59 + 6.22i)41-s + (−0.754 − 1.30i)43-s + 2.82·47-s + ⋯
L(s)  = 1  + (0.875 + 1.51i)5-s + (0.209 − 0.977i)7-s + (0.964 + 0.556i)11-s + (0.133 + 0.0769i)13-s + (−0.707 − 1.22i)17-s + (1.06 + 0.612i)19-s + (0.410 − 0.237i)23-s + (−1.03 + 1.78i)25-s + (−0.656 + 0.379i)29-s + 1.45i·31-s + (1.66 − 0.538i)35-s + (0.640 − 1.11i)37-s + (−0.561 + 0.971i)41-s + (−0.114 − 0.199i)43-s + 0.412·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.749 - 0.662i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.749 - 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78316 + 0.675154i\)
\(L(\frac12)\) \(\approx\) \(1.78316 + 0.675154i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-0.554 + 2.58i)T \)
good5 \( 1 + (-1.95 - 3.39i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.19 - 1.84i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.480 - 0.277i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.91 + 5.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.62 - 2.66i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.96 + 1.13i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.53 - 2.04i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.08iT - 31T^{2} \)
37 \( 1 + (-3.89 + 6.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.59 - 6.22i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.754 + 1.30i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + (-0.0415 + 0.0239i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.91T + 59T^{2} \)
61 \( 1 - 6.96iT - 61T^{2} \)
67 \( 1 - 1.17T + 67T^{2} \)
71 \( 1 + 6.71iT - 71T^{2} \)
73 \( 1 + (3.52 - 2.03i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 3.94T + 79T^{2} \)
83 \( 1 + (3.84 + 6.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.71 + 4.69i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.9 + 8.07i)T + (48.5 - 84.0i)T^{2} \)
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

−10.38168542391789962116034825519, −9.764842041320465213709932600721, −8.979919582155989272424367043874, −7.37324859233884806441465590218, −7.07736136409196691008269975852, −6.27071339402366587256026425091, −5.08054942527996245929427061206, −3.83560920190067272409513024952, −2.85000732133682975870618806484, −1.53793174818755783173863094328, 1.15834432924289629085731013947, 2.24967444658113704605549710671, 3.89219757928964133110756504349, 4.99993528191457023717768976096, 5.73369985275012058250097350947, 6.42009173062795031649159901645, 7.972146365631804912424080656616, 8.801359232719931062794444588189, 9.170779024134619544745178520999, 9.951349561853197959030593618091

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