↑

Properties

Label 2-756-189.5-c1-0-22
Degree $2$
Conductor $756$
Sign $-0.668 + 0.743i$
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  + (0.277 + 1.70i)3-s + (0.630 − 3.57i)5-s + (−2.41 + 1.08i)7-s + (−2.84 + 0.949i)9-s + (−5.37 + 0.947i)11-s + (3.07 − 3.66i)13-s + (6.28 + 0.0845i)15-s − 3.33·17-s − 5.54i·19-s + (−2.53 − 3.82i)21-s + (−4.97 + 5.92i)23-s + (−7.67 − 2.79i)25-s + (−2.41 − 4.60i)27-s + (−4.08 − 4.86i)29-s + (1.15 + 3.18i)31-s + ⋯
L(s)  = 1  + (0.160 + 0.987i)3-s + (0.281 − 1.59i)5-s + (−0.911 + 0.411i)7-s + (−0.948 + 0.316i)9-s + (−1.61 + 0.285i)11-s + (0.853 − 1.01i)13-s + (1.62 + 0.0218i)15-s − 0.807·17-s − 1.27i·19-s + (−0.552 − 0.833i)21-s + (−1.03 + 1.23i)23-s + (−1.53 − 0.558i)25-s + (−0.464 − 0.885i)27-s + (−0.758 − 0.903i)29-s + (0.207 + 0.571i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.668 + 0.743i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.668 + 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.180690 - 0.405402i\)
\(L(\frac12)\) \(\approx\) \(0.180690 - 0.405402i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.277 - 1.70i)T \)
7 \( 1 + (2.41 - 1.08i)T \)
good5 \( 1 + (-0.630 + 3.57i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (5.37 - 0.947i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (-3.07 + 3.66i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + 3.33T + 17T^{2} \)
19 \( 1 + 5.54iT - 19T^{2} \)
23 \( 1 + (4.97 - 5.92i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (4.08 + 4.86i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (-1.15 - 3.18i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (-0.690 - 1.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.74 - 3.98i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (1.78 + 0.650i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (5.57 + 2.02i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-6.01 + 3.47i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (8.58 + 7.20i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (0.712 - 1.95i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (-1.69 + 9.58i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (1.02 + 0.590i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.38 - 1.95i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.24 - 7.06i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-7.60 + 6.38i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + 6.67T + 89T^{2} \)
97 \( 1 + (3.89 - 10.7i)T + (-74.3 - 62.3i)T^{2} \)
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

−9.806469848596976910045871758456, −9.305619994954462866357760483588, −8.472187013026228797737059379546, −7.84472981965662445265950611225, −6.09666001726704033226433555596, −5.35792939881999448311377800151, −4.75078809251779586644532384760, −3.53006414035779561972824615602, −2.36525631888291090242657956549, −0.19960584974056098836031256754, 2.08667905844689310805673523589, 2.90578415632947915489969253194, 3.90303396437028075288189694034, 5.89738782700170917310657355953, 6.30566680998258506758623323925, 7.12619252866160498029018587680, 7.83273643256462823132427056189, 8.845432781003380102381136271325, 10.03727726886532949510953142162, 10.67906131677320474193582608446

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