↑

Properties

Label 2-154-77.10-c1-0-3
Degree $2$
Conductor $154$
Sign $0.442 + 0.896i$
Analytic cond. $1.22969$
Root an. cond. $1.10891$
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  + (−0.866 + 0.5i)2-s + (−0.889 − 0.513i)3-s + (0.499 − 0.866i)4-s + (1.09 − 0.629i)5-s + 1.02·6-s + (−2.11 − 1.59i)7-s + 0.999i·8-s + (−0.972 − 1.68i)9-s + (−0.629 + 1.09i)10-s + (1.85 − 2.74i)11-s + (−0.889 + 0.513i)12-s + 4.08·13-s + (2.62 + 0.319i)14-s − 1.29·15-s + (−0.5 − 0.866i)16-s + (1.60 − 2.77i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.513 − 0.296i)3-s + (0.249 − 0.433i)4-s + (0.487 − 0.281i)5-s + 0.419·6-s + (−0.799 − 0.600i)7-s + 0.353i·8-s + (−0.324 − 0.561i)9-s + (−0.199 + 0.344i)10-s + (0.560 − 0.828i)11-s + (−0.256 + 0.148i)12-s + 1.13·13-s + (0.701 + 0.0854i)14-s − 0.333·15-s + (−0.125 − 0.216i)16-s + (0.388 − 0.672i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.442 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.442 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $0.442 + 0.896i$
Analytic conductor: \(1.22969\)
Root analytic conductor: \(1.10891\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{154} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 154,\ (\ :1/2),\ 0.442 + 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.588727 - 0.366107i\)
\(L(\frac12)\) \(\approx\) \(0.588727 - 0.366107i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (2.11 + 1.59i)T \)
11 \( 1 + (-1.85 + 2.74i)T \)
good3 \( 1 + (0.889 + 0.513i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.09 + 0.629i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 4.08T + 13T^{2} \)
17 \( 1 + (-1.60 + 2.77i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.81 + 6.60i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.12 - 7.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.54iT - 29T^{2} \)
31 \( 1 + (7.95 + 4.59i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.154 - 0.266i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.05T + 41T^{2} \)
43 \( 1 - 7.57iT - 43T^{2} \)
47 \( 1 + (-4.07 + 2.35i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.39 + 4.15i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.36 - 1.36i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.755 - 1.30i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.69 - 2.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.50T + 71T^{2} \)
73 \( 1 + (0.483 - 0.837i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-13.5 + 7.80i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.32T + 83T^{2} \)
89 \( 1 + (-9.22 + 5.32i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.6iT - 97T^{2} \)
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

−13.01746980000339528488466764741, −11.44361981947165205045297807053, −10.93505909573956732098831088936, −9.333872836714921460112078831324, −9.022356111693316975766479335767, −7.30402997514095969609598983611, −6.38675151707025516463909672146, −5.54844878280316835724532214949, −3.47059241140345982646103329995, −0.921503479774768090490739097816, 2.17481583267637887393102449702, 3.92355891380488842702069707054, 5.78002143972810576662992242471, 6.56371723363167183645308156022, 8.209114174238582865100593912854, 9.179117935635721068513365251641, 10.33239654673329677038068878188, 10.76766710952862172660530307064, 12.16279906948912605275817536118, 12.75832934098393779422553188444

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