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Properties

Label 2-968-88.53-c1-0-54
Degree $2$
Conductor $968$
Sign $0.214 - 0.976i$
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  + (−0.784 + 1.17i)2-s + (0.866 + 1.19i)3-s + (−0.768 − 1.84i)4-s + (3.38 − 1.09i)5-s + (−2.08 + 0.0836i)6-s + (3.11 + 2.26i)7-s + (2.77 + 0.544i)8-s + (0.255 − 0.787i)9-s + (−1.36 + 4.84i)10-s + (1.53 − 2.51i)12-s + (−4.54 − 1.47i)13-s + (−5.10 + 1.88i)14-s + (4.24 + 3.08i)15-s + (−2.81 + 2.83i)16-s + (0.492 + 1.51i)17-s + (0.726 + 0.919i)18-s + ⋯
L(s)  = 1  + (−0.554 + 0.831i)2-s + (0.500 + 0.688i)3-s + (−0.384 − 0.923i)4-s + (1.51 − 0.491i)5-s + (−0.850 + 0.0341i)6-s + (1.17 + 0.855i)7-s + (0.981 + 0.192i)8-s + (0.0853 − 0.262i)9-s + (−0.430 + 1.53i)10-s + (0.443 − 0.726i)12-s + (−1.26 − 0.409i)13-s + (−1.36 + 0.504i)14-s + (1.09 + 0.795i)15-s + (−0.704 + 0.709i)16-s + (0.119 + 0.367i)17-s + (0.171 + 0.216i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.214 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.214 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.214 - 0.976i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ 0.214 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56525 + 1.25941i\)
\(L(\frac12)\) \(\approx\) \(1.56525 + 1.25941i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.784 - 1.17i)T \)
11 \( 1 \)
good3 \( 1 + (-0.866 - 1.19i)T + (-0.927 + 2.85i)T^{2} \)
5 \( 1 + (-3.38 + 1.09i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (-3.11 - 2.26i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (4.54 + 1.47i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.492 - 1.51i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.43 - 3.35i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + 3.24T + 23T^{2} \)
29 \( 1 + (-4.34 + 5.97i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.10 - 3.41i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.358 + 0.493i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (7.52 - 5.46i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 3.06iT - 43T^{2} \)
47 \( 1 + (-3.90 + 2.83i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.96 - 1.28i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.31 + 4.56i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.05 + 0.668i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 1.47iT - 67T^{2} \)
71 \( 1 + (-3.21 - 9.90i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (7.52 + 5.46i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.985 - 3.03i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (11.9 - 3.89i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 + (5.29 - 16.3i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.923438246607355637477308221378, −9.403488535806466379178281307541, −8.515637221382066600881297407264, −8.073900762704432005779826322047, −6.76478705594007291115616377339, −5.66191509706478064598510070210, −5.28184258681971038686185284044, −4.35570657801743720056933795188, −2.47975369037669390319474537169, −1.47337079779561935885782150526, 1.32006405316255237483445763538, 2.11451043744753889450939825740, 2.84258422538804622430470872499, 4.48764912958082711141533049201, 5.28459863112459587466977533748, 6.99585958655399933639254325538, 7.27471496292290851911788395732, 8.233082532569784133523511717918, 9.132557241455970813257754211303, 9.999050585425518636657660258542

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