↑

Properties

Label 2-936-117.94-c1-0-38
Degree $2$
Conductor $936$
Sign $-0.988 + 0.153i$
Analytic cond. $7.47399$
Root an. cond. $2.73386$
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  + (−1.70 − 0.293i)3-s + (2.01 − 3.49i)5-s − 3.11·7-s + (2.82 + 1.00i)9-s + (1.33 − 2.31i)11-s + (−3.21 − 1.63i)13-s + (−4.46 + 5.36i)15-s + (3.16 − 5.47i)17-s + (−2.62 + 4.53i)19-s + (5.31 + 0.913i)21-s + 3.40·23-s + (−5.62 − 9.74i)25-s + (−4.53 − 2.53i)27-s + (−4.30 + 7.45i)29-s + (2.18 − 3.79i)31-s + ⋯
L(s)  = 1  + (−0.985 − 0.169i)3-s + (0.901 − 1.56i)5-s − 1.17·7-s + (0.942 + 0.333i)9-s + (0.403 − 0.699i)11-s + (−0.890 − 0.454i)13-s + (−1.15 + 1.38i)15-s + (0.766 − 1.32i)17-s + (−0.601 + 1.04i)19-s + (1.16 + 0.199i)21-s + 0.710·23-s + (−1.12 − 1.94i)25-s + (−0.872 − 0.488i)27-s + (−0.799 + 1.38i)29-s + (0.393 − 0.680i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.988 + 0.153i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (913, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.988 + 0.153i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0538793 - 0.697441i\)
\(L(\frac12)\) \(\approx\) \(0.0538793 - 0.697441i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 + 0.293i)T \)
13 \( 1 + (3.21 + 1.63i)T \)
good5 \( 1 + (-2.01 + 3.49i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 3.11T + 7T^{2} \)
11 \( 1 + (-1.33 + 2.31i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.16 + 5.47i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.62 - 4.53i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.40T + 23T^{2} \)
29 \( 1 + (4.30 - 7.45i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.18 + 3.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.81 - 6.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.10T + 41T^{2} \)
43 \( 1 + 4.85T + 43T^{2} \)
47 \( 1 + (1.83 + 3.18i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.70T + 53T^{2} \)
59 \( 1 + (1.17 + 2.02i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 7.95T + 61T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 + (-4.55 + 7.88i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.32T + 73T^{2} \)
79 \( 1 + (-0.0416 - 0.0721i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.29 - 3.97i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.92 + 6.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.75T + 97T^{2} \)
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

−9.779370388231572156581472064831, −9.071305247045778070748544233904, −8.011339311727795806801158848126, −6.86346780433928528862135316280, −6.03229536128463572006735856671, −5.34811394623302158114113212747, −4.69251307350932494077423611500, −3.21377039556852531557455922162, −1.53312519955026052673310533339, −0.36915002096000076740784934675, 1.95455478506723780720618161482, 3.10817472828056815045754529501, 4.21835436335733758654523675575, 5.49851600775084886245854433663, 6.44539532112593275879622594976, 6.65851139207017542925507650211, 7.49868188940173361591118282972, 9.311806646648536250095855594794, 9.826909032597859192648307344046, 10.35014410095129658694190389009

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