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Properties

Label 2-968-88.5-c1-0-27
Degree $2$
Conductor $968$
Sign $-0.662 - 0.748i$
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  + (−1.37 − 0.330i)2-s + (−1.38 + 1.90i)3-s + (1.78 + 0.908i)4-s + (3.96 + 1.28i)5-s + (2.53 − 2.16i)6-s + (0.754 − 0.548i)7-s + (−2.14 − 1.83i)8-s + (−0.783 − 2.41i)9-s + (−5.02 − 3.08i)10-s + (−4.19 + 2.13i)12-s + (−2.78 + 0.906i)13-s + (−1.21 + 0.504i)14-s + (−7.93 + 5.76i)15-s + (2.34 + 3.23i)16-s + (−0.755 + 2.32i)17-s + (0.280 + 3.57i)18-s + ⋯
L(s)  = 1  + (−0.972 − 0.233i)2-s + (−0.798 + 1.09i)3-s + (0.890 + 0.454i)4-s + (1.77 + 0.575i)5-s + (1.03 − 0.882i)6-s + (0.285 − 0.207i)7-s + (−0.760 − 0.649i)8-s + (−0.261 − 0.804i)9-s + (−1.58 − 0.974i)10-s + (−1.21 + 0.616i)12-s + (−0.773 + 0.251i)13-s + (−0.325 + 0.134i)14-s + (−2.04 + 1.48i)15-s + (0.587 + 0.809i)16-s + (−0.183 + 0.564i)17-s + (0.0662 + 0.842i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 - 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $-0.662 - 0.748i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ -0.662 - 0.748i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.366929 + 0.814693i\)
\(L(\frac12)\) \(\approx\) \(0.366929 + 0.814693i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 + 0.330i)T \)
11 \( 1 \)
good3 \( 1 + (1.38 - 1.90i)T + (-0.927 - 2.85i)T^{2} \)
5 \( 1 + (-3.96 - 1.28i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.754 + 0.548i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (2.78 - 0.906i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.755 - 2.32i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.57 - 2.17i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + 3.47T + 23T^{2} \)
29 \( 1 + (-2.68 - 3.69i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.13 + 3.47i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.66 + 3.67i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.33 - 2.42i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 11.4iT - 43T^{2} \)
47 \( 1 + (2.63 + 1.91i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.618 + 0.200i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-1.71 - 2.35i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-10.1 - 3.30i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 5.42iT - 67T^{2} \)
71 \( 1 + (2.39 - 7.37i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-10.5 + 7.67i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.49 + 4.59i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.627 - 0.203i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + 2.74T + 89T^{2} \)
97 \( 1 + (-0.565 - 1.73i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.23438606963461819028910535901, −9.736100494194554112321466450484, −9.138571474291314892768834607907, −7.942534261942813513032110191726, −6.76259946632224050024683797899, −6.07847044462710051436657670791, −5.31821276708793236830928731463, −4.10338967374603196717284378020, −2.66691184109772253213629045299, −1.65838537086886284476731634679, 0.60587465532577180464264649341, 1.80489541957878857989656709850, 2.42027240981208179538632994679, 5.10353185286490930407267202134, 5.56611188586946606605557288196, 6.49540522480057590015493208788, 6.94825209869219444028688881936, 8.075725912576276023729882743552, 8.911473755775123205084694115086, 9.713529189513693998679251657585

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