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Properties

Label 2-966-7.2-c1-0-17
Degree $2$
Conductor $966$
Sign $0.754 + 0.655i$
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 + (−0.5 + 0.866i)5-s + 0.999·6-s + (1.85 − 1.88i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (0.499 + 0.866i)10-s + (0.202 + 0.350i)11-s + (0.499 − 0.866i)12-s + 4.12·13-s + (−0.702 − 2.55i)14-s − 0.999·15-s + (−0.5 + 0.866i)16-s + (−2.65 − 4.59i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s + 0.408·6-s + (0.702 − 0.711i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 + 0.273i)10-s + (0.0609 + 0.105i)11-s + (0.144 − 0.249i)12-s + 1.14·13-s + (−0.187 − 0.681i)14-s − 0.258·15-s + (−0.125 + 0.216i)16-s + (−0.644 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.754 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.754 + 0.655i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.754 + 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.06410 - 0.771277i\)
\(L(\frac12)\) \(\approx\) \(2.06410 - 0.771277i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.85 + 1.88i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.202 - 0.350i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.12T + 13T^{2} \)
17 \( 1 + (2.65 + 4.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.35 + 2.35i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 9.12T + 29T^{2} \)
31 \( 1 + (-4.91 - 8.51i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.95 + 6.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.7T + 41T^{2} \)
43 \( 1 - 0.808T + 43T^{2} \)
47 \( 1 + (-5.41 + 9.38i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.0955 - 0.165i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.15 - 7.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.41 + 2.45i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.92T + 71T^{2} \)
73 \( 1 + (2.29 + 3.97i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.96 + 8.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.40T + 83T^{2} \)
89 \( 1 + (3.95 - 6.84i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 15.3T + 97T^{2} \)
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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

−10.35426733365625387933183238731, −9.038040360129181721564180345702, −8.553817611764915884635694330415, −7.33972770102314063642244741880, −6.57791988653479721618687218261, −5.16908208706956117697784065053, −4.52264858475533961490706147186, −3.56222591461921974171309353028, −2.65755195892137485364228377719, −1.10986971790949455538385014326, 1.34310429369258863994140254302, 2.74532695135645841138657585530, 4.02239022172035898735972092187, 4.86464688344343991132537336350, 6.08059828597294100251896222130, 6.43409896484086618321391790559, 7.84982127265366763383842442756, 8.364763861450404523022245686988, 8.762398773574373561994883583583, 9.999004970813959343818517125239

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