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Properties

Label 2-968-88.5-c1-0-84
Degree $2$
Conductor $968$
Sign $0.999 - 0.0373i$
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  + (0.301 + 1.38i)2-s + (1.50 − 2.06i)3-s + (−1.81 + 0.832i)4-s + (2.56 + 0.832i)5-s + (3.30 + 1.45i)6-s + (3.19 − 2.32i)7-s + (−1.69 − 2.26i)8-s + (−1.08 − 3.35i)9-s + (−0.379 + 3.79i)10-s + (−1.01 + 5.00i)12-s + (−0.416 + 0.135i)13-s + (4.16 + 3.71i)14-s + (5.57 − 4.04i)15-s + (2.61 − 3.02i)16-s + (0.739 − 2.27i)17-s + (4.30 − 2.51i)18-s + ⋯
L(s)  = 1  + (0.212 + 0.977i)2-s + (0.867 − 1.19i)3-s + (−0.909 + 0.416i)4-s + (1.14 + 0.372i)5-s + (1.35 + 0.593i)6-s + (1.20 − 0.877i)7-s + (−0.600 − 0.799i)8-s + (−0.363 − 1.11i)9-s + (−0.119 + 1.19i)10-s + (−0.291 + 1.44i)12-s + (−0.115 + 0.0375i)13-s + (1.11 + 0.993i)14-s + (1.43 − 1.04i)15-s + (0.653 − 0.756i)16-s + (0.179 − 0.552i)17-s + (1.01 − 0.593i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.999 - 0.0373i$
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: \((2,\ 968,\ (\ :1/2),\ 0.999 - 0.0373i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.83158 + 0.0528372i\)
\(L(\frac12)\) \(\approx\) \(2.83158 + 0.0528372i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.301 - 1.38i)T \)
11 \( 1 \)
good3 \( 1 + (-1.50 + 2.06i)T + (-0.927 - 2.85i)T^{2} \)
5 \( 1 + (-2.56 - 0.832i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (-3.19 + 2.32i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.416 - 0.135i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.739 + 2.27i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.14 - 4.32i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + 1.35T + 23T^{2} \)
29 \( 1 + (-0.0866 - 0.119i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.489 - 1.50i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.33 + 5.97i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (2.92 + 2.12i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 6.62iT - 43T^{2} \)
47 \( 1 + (2.44 + 1.77i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (4.41 - 1.43i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-7.35 - 10.1i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.627 - 0.203i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 3.00iT - 67T^{2} \)
71 \( 1 + (3.26 - 10.0i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-8.61 + 6.25i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.99 - 6.13i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.17 + 0.705i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 + (-2.68 - 8.25i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.835813786920975478722577558203, −8.805469408912798664704181928111, −8.105604427372673008460351592572, −7.47357594477401157595230180057, −6.80719339899136729432813943992, −5.96416676849488658275637841703, −4.95819643291526828801356298986, −3.76817065838731964426169287029, −2.37604113480691310551152175981, −1.33626667343194804875101508290, 1.77231736825601048729061575442, 2.45026527461609864847138821998, 3.61164056641655881460520104704, 4.80760385299111195004796608425, 5.08154139300808654959361103798, 6.18319005284047350046280643296, 8.189428021541295488657516650099, 8.667376364818201047649156471127, 9.313046987954428372891966490218, 9.959403676084569457304383592453

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