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Properties

Label 2-968-11.3-c1-0-2
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  + (−0.482 − 1.48i)3-s + (−2.88 − 2.09i)5-s + (−0.965 + 2.97i)7-s + (0.454 − 0.330i)9-s + (−4.14 + 3.01i)13-s + (−1.71 + 5.28i)15-s + (1.61 + 1.17i)17-s + (1.23 + 3.80i)19-s + 4.87·21-s + 2.43·23-s + (2.37 + 7.30i)25-s + (−4.49 − 3.26i)27-s + (1.58 − 4.87i)29-s + (4.49 − 3.26i)31-s + (8.99 − 6.53i)35-s + ⋯
L(s)  = 1  + (−0.278 − 0.857i)3-s + (−1.28 − 0.936i)5-s + (−0.364 + 1.12i)7-s + (0.151 − 0.110i)9-s + (−1.14 + 0.835i)13-s + (−0.443 + 1.36i)15-s + (0.392 + 0.285i)17-s + (0.283 + 0.872i)19-s + 1.06·21-s + 0.508·23-s + (0.474 + 1.46i)25-s + (−0.865 − 0.629i)27-s + (0.293 − 0.904i)29-s + (0.808 − 0.587i)31-s + (1.52 − 1.10i)35-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

\(L(1)\) \(\approx\) \(0.669407 + 0.233894i\)
\(L(\frac12)\) \(\approx\) \(0.669407 + 0.233894i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.482 + 1.48i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (2.88 + 2.09i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.965 - 2.97i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (4.14 - 3.01i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.61 - 1.17i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.23 - 3.80i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 2.43T + 23T^{2} \)
29 \( 1 + (-1.58 + 4.87i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.49 + 3.26i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.33 - 7.19i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.347 - 1.06i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.12T + 43T^{2} \)
47 \( 1 + (-2.47 - 7.60i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (9.90 - 7.19i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.41 + 7.42i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.908 - 0.660i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 9.56T + 67T^{2} \)
71 \( 1 + (-7.02 - 5.10i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.58 - 4.87i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (8.99 - 6.53i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.709 - 0.515i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 2.68T + 89T^{2} \)
97 \( 1 + (12.5 - 9.14i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.832703577901953772446593540117, −9.261383528102800094972082956467, −8.191479630106169849684098372056, −7.74357230049674014023182206000, −6.76297759829433578346609810561, −5.88360862131935933162356626282, −4.81965730501724992104043739156, −3.95528154062837327568856908587, −2.54318486927288227240168104634, −1.12609283814508397222872192531, 0.40602369501207678090111593996, 2.89411778344319084791083788264, 3.64420321355245485868387036150, 4.50659900512846252396051793059, 5.30345502434784498724312402347, 7.03091283502900566298707724656, 7.14905740766578045637419222274, 8.037096497261317012836499617702, 9.344294054181034321634120370136, 10.23834838613389339010293433143

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