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Properties

Label 2-968-88.35-c1-0-1
Degree $2$
Conductor $968$
Sign $-0.909 - 0.416i$
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  + (−1.34 − 0.437i)2-s + (−0.0359 + 0.0261i)3-s + (1.61 + 1.17i)4-s + (0.0598 − 0.0194i)6-s + (−1.66 − 2.28i)8-s + (−0.926 + 2.85i)9-s − 0.0889·12-s + (1.23 + 3.80i)16-s + (−3.76 + 1.22i)17-s + (2.49 − 3.43i)18-s + (−2.65 − 3.66i)19-s + (0.119 + 0.0388i)24-s + (−4.04 + 2.93i)25-s + (−0.0824 − 0.253i)27-s − 5.65i·32-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (−0.0207 + 0.0150i)3-s + (0.809 + 0.587i)4-s + (0.0244 − 0.00793i)6-s + (−0.587 − 0.809i)8-s + (−0.308 + 0.950i)9-s − 0.0256·12-s + (0.309 + 0.951i)16-s + (−0.913 + 0.296i)17-s + (0.587 − 0.808i)18-s + (−0.610 − 0.839i)19-s + (0.0244 + 0.00793i)24-s + (−0.809 + 0.587i)25-s + (−0.0158 − 0.0488i)27-s − 1.00i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 - 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $-0.909 - 0.416i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ -0.909 - 0.416i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0381804 + 0.175068i\)
\(L(\frac12)\) \(\approx\) \(0.0381804 + 0.175068i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 + 0.437i)T \)
11 \( 1 \)
good3 \( 1 + (0.0359 - 0.0261i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (3.76 - 1.22i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.65 + 3.66i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (7.45 + 10.2i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 - 12.7iT - 43T^{2} \)
47 \( 1 + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (9.38 + 6.81i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 12.3T + 67T^{2} \)
71 \( 1 + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (7.37 - 10.1i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (12.2 - 3.97i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + 17.8T + 89T^{2} \)
97 \( 1 + (5.58 - 17.1i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.37530492563879473086764010716, −9.544209524046389047759754334694, −8.675746660451813013180675088032, −8.095241886814141247249405871781, −7.16099410842319220092261860553, −6.36796155191010471947336625734, −5.19068869853605385350088027700, −3.99609730131615657956111421278, −2.70595147384849362482474060645, −1.78583644548192812089567404171, 0.10649088180158756138715827104, 1.71939691483490035977783903654, 2.99654650048010015027728634458, 4.31217694260404641171611152605, 5.68058520119245833170905205859, 6.37345876487170036472002987761, 7.12159486820145165426833879927, 8.183810376307093680571869611593, 8.768043636578451530712067860290, 9.604410166728345718024155946245

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