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Properties

Label 2-968-88.37-c1-0-21
Degree $2$
Conductor $968$
Sign $-0.756 - 0.654i$
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  + (1.22 + 0.713i)2-s + (−2.42 − 0.789i)3-s + (0.982 + 1.74i)4-s + (1.58 + 2.18i)5-s + (−2.40 − 2.69i)6-s + (1.22 + 3.75i)7-s + (−0.0433 + 2.82i)8-s + (2.85 + 2.07i)9-s + (0.379 + 3.79i)10-s + (−1.01 − 5.00i)12-s + (0.257 − 0.354i)13-s + (−1.18 + 5.45i)14-s + (−2.12 − 6.54i)15-s + (−2.07 + 3.42i)16-s + (1.93 − 1.40i)17-s + (2.00 + 4.56i)18-s + ⋯
L(s)  = 1  + (0.863 + 0.504i)2-s + (−1.40 − 0.455i)3-s + (0.491 + 0.871i)4-s + (0.708 + 0.975i)5-s + (−0.981 − 1.10i)6-s + (0.461 + 1.41i)7-s + (−0.0153 + 0.999i)8-s + (0.951 + 0.691i)9-s + (0.119 + 1.19i)10-s + (−0.291 − 1.44i)12-s + (0.0713 − 0.0982i)13-s + (−0.317 + 1.45i)14-s + (−0.549 − 1.69i)15-s + (−0.517 + 0.855i)16-s + (0.469 − 0.341i)17-s + (0.472 + 1.07i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $-0.756 - 0.654i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ -0.756 - 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.608383 + 1.63317i\)
\(L(\frac12)\) \(\approx\) \(0.608383 + 1.63317i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.713i)T \)
11 \( 1 \)
good3 \( 1 + (2.42 + 0.789i)T + (2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.58 - 2.18i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.22 - 3.75i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-0.257 + 0.354i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.93 + 1.40i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (5.08 + 1.65i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.35T + 23T^{2} \)
29 \( 1 + (-0.140 + 0.0455i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.28 + 0.930i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-7.01 + 2.28i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.11 - 3.43i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 6.62iT - 43T^{2} \)
47 \( 1 + (-0.934 + 2.87i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.72 - 3.75i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (11.8 - 3.86i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.387 + 0.533i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 3.00iT - 67T^{2} \)
71 \( 1 + (-8.54 + 6.21i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.28 - 10.1i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.21 - 3.78i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.34 - 1.84i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 + (7.02 + 5.10i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.88224961610867506295782503003, −9.587676701489841774547026453709, −8.428705054860964492542215486390, −7.45427009474919308130569816117, −6.41446623305507395378967354464, −6.12934525001726158541639535669, −5.42770686165646456412780506647, −4.58124256337726119343668924587, −2.90522867444843309620246403596, −2.01733292491031301247192178102, 0.72680578717857335392673754942, 1.76650151122299526391129216595, 3.80265968247592824237121675888, 4.52560289638881843729477052197, 5.14111322774312059184993692802, 5.97548929147325241182995332065, 6.65711802504736073735890744126, 7.88778207349864784224119748628, 9.268141105260269601318104169590, 10.18175484277395471578561301113

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