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Properties

Label 2-966-161.160-c3-0-95
Degree $2$
Conductor $966$
Sign $-0.535 - 0.844i$
Analytic cond. $56.9958$
Root an. cond. $7.54955$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3i·3-s + 4·4-s − 18.7·5-s − 6i·6-s + (2.04 − 18.4i)7-s + 8·8-s − 9·9-s − 37.4·10-s − 52.4i·11-s − 12i·12-s − 2.52i·13-s + (4.08 − 36.8i)14-s + 56.2i·15-s + 16·16-s + 28.4·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577i·3-s + 0.5·4-s − 1.67·5-s − 0.408i·6-s + (0.110 − 0.993i)7-s + 0.353·8-s − 0.333·9-s − 1.18·10-s − 1.43i·11-s − 0.288i·12-s − 0.0538i·13-s + (0.0780 − 0.702i)14-s + 0.968i·15-s + 0.250·16-s + 0.406·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.535 - 0.844i$
Analytic conductor: \(56.9958\)
Root analytic conductor: \(7.54955\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :3/2),\ -0.535 - 0.844i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4911514998\)
\(L(\frac12)\) \(\approx\) \(0.4911514998\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
3 \( 1 + 3iT \)
7 \( 1 + (-2.04 + 18.4i)T \)
23 \( 1 + (68.9 + 86.0i)T \)
good5 \( 1 + 18.7T + 125T^{2} \)
11 \( 1 + 52.4iT - 1.33e3T^{2} \)
13 \( 1 + 2.52iT - 2.19e3T^{2} \)
17 \( 1 - 28.4T + 4.91e3T^{2} \)
19 \( 1 + 57.9T + 6.85e3T^{2} \)
29 \( 1 + 33.3T + 2.43e4T^{2} \)
31 \( 1 - 142. iT - 2.97e4T^{2} \)
37 \( 1 + 98.3iT - 5.06e4T^{2} \)
41 \( 1 - 121. iT - 6.89e4T^{2} \)
43 \( 1 + 178. iT - 7.95e4T^{2} \)
47 \( 1 + 23.5iT - 1.03e5T^{2} \)
53 \( 1 - 558. iT - 1.48e5T^{2} \)
59 \( 1 - 61.3iT - 2.05e5T^{2} \)
61 \( 1 - 348.T + 2.26e5T^{2} \)
67 \( 1 - 150. iT - 3.00e5T^{2} \)
71 \( 1 - 31.1T + 3.57e5T^{2} \)
73 \( 1 - 448. iT - 3.89e5T^{2} \)
79 \( 1 - 1.14e3iT - 4.93e5T^{2} \)
83 \( 1 + 1.31e3T + 5.71e5T^{2} \)
89 \( 1 + 1.52e3T + 7.04e5T^{2} \)
97 \( 1 + 687.T + 9.12e5T^{2} \)
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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

−8.597381517609277586581123008916, −8.140435172108094494349737857067, −7.33088246751743661928441139224, −6.65708910619944824104557293289, −5.60358651170320785317961516039, −4.34785819594740413351064329230, −3.76219875594125862538909449349, −2.87115980294853648079099272162, −1.07849679590768003837698729164, −0.10760348856658528396774816152, 1.99353554821478905536046814114, 3.19956368676714653840608262307, 4.11042227385942767414717712735, 4.69740996219838002481392045398, 5.65108068353045666124172957588, 6.81607807800886746839032684487, 7.71276944007698640733000687261, 8.331582306080560656296161539244, 9.405141685555909524854334610465, 10.24716313391635765362084715175

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