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Properties

Label 2-968-11.9-c1-0-18
Degree $2$
Conductor $968$
Sign $0.649 + 0.760i$
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  + (0.592 − 0.430i)3-s + (−0.0828 − 0.254i)5-s + (−0.592 − 0.430i)7-s + (−0.761 + 2.34i)9-s + (0.761 − 2.34i)13-s + (−0.158 − 0.115i)15-s + (−1.77 − 5.45i)17-s + (5.44 − 3.95i)19-s − 0.535·21-s + 8.19·23-s + (3.98 − 2.89i)25-s + (1.23 + 3.80i)27-s + (1.99 + 1.44i)29-s + (−0.391 + 1.20i)31-s + (−0.0606 + 0.186i)35-s + ⋯
L(s)  = 1  + (0.341 − 0.248i)3-s + (−0.0370 − 0.113i)5-s + (−0.223 − 0.162i)7-s + (−0.253 + 0.781i)9-s + (0.211 − 0.649i)13-s + (−0.0409 − 0.0297i)15-s + (−0.429 − 1.32i)17-s + (1.24 − 0.907i)19-s − 0.116·21-s + 1.70·23-s + (0.797 − 0.579i)25-s + (0.237 + 0.732i)27-s + (0.370 + 0.268i)29-s + (−0.0703 + 0.216i)31-s + (−0.0102 + 0.0315i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.649 + 0.760i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ 0.649 + 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55126 - 0.715216i\)
\(L(\frac12)\) \(\approx\) \(1.55126 - 0.715216i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.592 + 0.430i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.0828 + 0.254i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (0.592 + 0.430i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.761 + 2.34i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.77 + 5.45i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-5.44 + 3.95i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 8.19T + 23T^{2} \)
29 \( 1 + (-1.99 - 1.44i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.391 - 1.20i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.83 + 1.33i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.45 + 2.50i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (-5.44 + 3.95i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.84 - 8.74i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.28 + 3.84i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.16 + 12.8i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 10.1T + 67T^{2} \)
71 \( 1 + (-2.92 - 9.00i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.98 - 2.89i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (4.50 - 13.8i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.53 - 7.79i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 9.53T + 89T^{2} \)
97 \( 1 + (-0.595 + 1.83i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.793434898682762400774490772785, −9.001764238782961348360453301946, −8.295430265061630549821821356763, −7.26289550608149241714362509009, −6.82016084521287489861181308320, −5.26804822497353461807455187233, −4.89085433005637196535834420223, −3.25192931853281554800248840966, −2.57818174678601953544620101405, −0.873940711884037358743993440492, 1.36237519962995145205507787158, 2.96503893251976022381080588437, 3.67542235262123895417193746241, 4.78627916529442796374579963226, 5.96193319578195878476723113817, 6.65022678141919875233429712378, 7.65392324294065940326195433866, 8.709035570054014325196524905010, 9.179773646117994057140051291574, 10.02381576743999499697815180816

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