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Properties

Label 2-966-7.4-c1-0-13
Degree $2$
Conductor $966$
Sign $0.991 - 0.126i$
Analytic cond. $7.71354$
Root an. cond. $2.77732$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−1.5 − 2.59i)5-s − 0.999·6-s + (0.5 + 2.59i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (1.5 − 2.59i)10-s + (2.5 − 4.33i)11-s + (−0.499 − 0.866i)12-s + (−2 + 1.73i)14-s + 3·15-s + (−0.5 − 0.866i)16-s + (2 − 3.46i)17-s + (0.499 − 0.866i)18-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.670 − 1.16i)5-s − 0.408·6-s + (0.188 + 0.981i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.474 − 0.821i)10-s + (0.753 − 1.30i)11-s + (−0.144 − 0.249i)12-s + (−0.534 + 0.462i)14-s + 0.774·15-s + (−0.125 − 0.216i)16-s + (0.485 − 0.840i)17-s + (0.117 − 0.204i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.991 - 0.126i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45931 + 0.0926083i\)
\(L(\frac12)\) \(\approx\) \(1.45931 + 0.0926083i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 2.59i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (-4.5 + 7.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6 - 10.3i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.5 - 4.33i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-7 + 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 17T + 83T^{2} \)
89 \( 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 13T + 97T^{2} \)
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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

−9.650407768669745372666202718997, −9.087920251240340788560753482955, −8.357840045885784651778141879406, −7.74471412427014847447529700448, −6.25770126867669343429437759305, −5.70981443157699516037835041513, −4.79171287602688970636966379969, −4.08377005163099850398828949352, −2.91819436233278651467542769673, −0.75326052828357931026428979232, 1.24029064146650868114045456993, 2.53069887300620699468188616683, 3.81663452723781749276112096808, 4.30963973791060101563727783530, 5.72125152001093631461818901784, 6.82450806400615382653206040504, 7.24599006001878626005770668970, 8.088741524543364875753841812108, 9.495037505427235443974712315924, 10.25261162515689573400733939698

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