↑

Properties

Label 2-968-88.19-c1-0-43
Degree $2$
Conductor $968$
Sign $0.958 - 0.286i$
Analytic cond. $7.72951$
Root an. cond. $2.78020$
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  + (−0.831 − 1.14i)2-s + (0.618 + 1.90i)3-s + (−0.618 + 1.90i)4-s + (1.66 − 2.28i)6-s + (2.68 − 0.874i)8-s + (−0.809 + 0.587i)9-s − 3.99·12-s + (−3.23 − 2.35i)16-s + (3.32 − 4.57i)17-s + (1.34 + 0.437i)18-s + (8.06 − 2.62i)19-s + (3.32 + 4.57i)24-s + (1.54 + 4.75i)25-s + (3.23 + 2.35i)27-s + 5.65i·32-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (0.356 + 1.09i)3-s + (−0.309 + 0.951i)4-s + (0.678 − 0.934i)6-s + (0.951 − 0.309i)8-s + (−0.269 + 0.195i)9-s − 1.15·12-s + (−0.809 − 0.587i)16-s + (0.806 − 1.10i)17-s + (0.317 + 0.103i)18-s + (1.85 − 0.601i)19-s + (0.678 + 0.934i)24-s + (0.309 + 0.951i)25-s + (0.622 + 0.452i)27-s + 0.999i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.286i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.958 - 0.286i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (723, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ 0.958 - 0.286i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38023 + 0.201819i\)
\(L(\frac12)\) \(\approx\) \(1.38023 + 0.201819i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.831 + 1.14i)T \)
11 \( 1 \)
good3 \( 1 + (-0.618 - 1.90i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.32 + 4.57i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-8.06 + 2.62i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (10.7 - 3.49i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 - 8.48iT - 43T^{2} \)
47 \( 1 + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.85 - 5.70i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 14T + 67T^{2} \)
71 \( 1 + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (16.1 + 5.24i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.66 + 2.28i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 18T + 89T^{2} \)
97 \( 1 + (8.09 - 5.87i)T + (29.9 - 92.2i)T^{2} \)
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

−9.812485814054689917506477541590, −9.510776758998169859136161456581, −8.795900215282151910972457928209, −7.71535688597177114058285562389, −7.02642726856748999922570370698, −5.29938131556421681876797322014, −4.60747256774488109939658552174, −3.39137474446855105255960182977, −2.94581417892924263598829772378, −1.15886717318384733764948724713, 0.999877725211212414818740559137, 2.02435868604398995908738758780, 3.60187525345255533036264675113, 5.08060657767811712694530541479, 5.92364350865537645902763071413, 6.83250805199909928764923346610, 7.49789368704775662339885992421, 8.160169577003454342218514473197, 8.793239240651029431098835156546, 9.983306128581646517250115318379

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