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Properties

Label 2-968-11.3-c1-0-21
Degree $2$
Conductor $968$
Sign $-0.782 + 0.622i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.927 − 2.85i)3-s + (2.42 + 1.76i)5-s + (0.618 − 1.90i)7-s + (−4.85 + 3.52i)9-s + (2.78 − 8.55i)15-s + (−4.85 − 3.52i)17-s + (−1.23 − 3.80i)19-s − 6·21-s + 23-s + (1.23 + 3.80i)25-s + (7.28 + 5.29i)27-s + (2.47 − 7.60i)29-s + (5.66 − 4.11i)31-s + (4.85 − 3.52i)35-s + (−0.309 + 0.951i)37-s + ⋯
L(s)  = 1  + (−0.535 − 1.64i)3-s + (1.08 + 0.788i)5-s + (0.233 − 0.718i)7-s + (−1.61 + 1.17i)9-s + (0.718 − 2.21i)15-s + (−1.17 − 0.855i)17-s + (−0.283 − 0.872i)19-s − 1.30·21-s + 0.208·23-s + (0.247 + 0.760i)25-s + (1.40 + 1.01i)27-s + (0.459 − 1.41i)29-s + (1.01 − 0.738i)31-s + (0.820 − 0.596i)35-s + (−0.0508 + 0.156i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $-0.782 + 0.622i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ -0.782 + 0.622i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.432776 - 1.23860i\)
\(L(\frac12)\) \(\approx\) \(0.432776 - 1.23860i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.927 + 2.85i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-2.42 - 1.76i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.618 + 1.90i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (4.85 + 3.52i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.23 + 3.80i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - T + 23T^{2} \)
29 \( 1 + (-2.47 + 7.60i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.66 + 4.11i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.23 + 3.80i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + (2.47 + 7.60i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.61 - 1.17i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.309 - 0.951i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.23 - 2.35i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 5T + 67T^{2} \)
71 \( 1 + (2.42 + 1.76i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.94 - 15.2i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-1.61 + 1.17i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.61 + 1.17i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 + (-5.66 + 4.11i)T + (29.9 - 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.842156862307999212612778440636, −8.706714601292794737064892053226, −7.71912206143910000242631447536, −6.85254356050670262244959977138, −6.58483533942736339458593338776, −5.69264575702249144944728666576, −4.55404179485391781129437347519, −2.68868216983682582746808199439, −2.03830571311696940907735680193, −0.63561363980084285658945317226, 1.74198883001627299499788086606, 3.21485037047170936623488307515, 4.49165149735128573401100900865, 5.01060340559596666817224087118, 5.81998825288188653518772211395, 6.47447277571424752090507794830, 8.400348372336118543109742936646, 8.859833007302295263036935976027, 9.555075583068005166271632819085, 10.31391518915341632003352781629

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