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Properties

Label 2-946-11.3-c1-0-12
Degree $2$
Conductor $946$
Sign $0.102 - 0.994i$
Analytic cond. $7.55384$
Root an. cond. $2.74842$
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  + (−0.809 + 0.587i)2-s + (0.536 + 1.65i)3-s + (0.309 − 0.951i)4-s + (1.03 + 0.750i)5-s + (−1.40 − 1.02i)6-s + (−0.0591 + 0.182i)7-s + (0.309 + 0.951i)8-s + (−0.0105 + 0.00767i)9-s − 1.27·10-s + (3.06 + 1.27i)11-s + 1.73·12-s + (3.94 − 2.86i)13-s + (−0.0591 − 0.182i)14-s + (−0.684 + 2.10i)15-s + (−0.809 − 0.587i)16-s + (−2.05 − 1.49i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.309 + 0.953i)3-s + (0.154 − 0.475i)4-s + (0.461 + 0.335i)5-s + (−0.573 − 0.416i)6-s + (−0.0223 + 0.0688i)7-s + (0.109 + 0.336i)8-s + (−0.00351 + 0.00255i)9-s − 0.403·10-s + (0.923 + 0.383i)11-s + 0.501·12-s + (1.09 − 0.794i)13-s + (−0.0158 − 0.0486i)14-s + (−0.176 + 0.543i)15-s + (−0.202 − 0.146i)16-s + (−0.498 − 0.361i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(946\)    =    \(2 \cdot 11 \cdot 43\)
Sign: $0.102 - 0.994i$
Analytic conductor: \(7.55384\)
Root analytic conductor: \(2.74842\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{946} (861, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 946,\ (\ :1/2),\ 0.102 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22678 + 1.10641i\)
\(L(\frac12)\) \(\approx\) \(1.22678 + 1.10641i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (-3.06 - 1.27i)T \)
43 \( 1 + T \)
good3 \( 1 + (-0.536 - 1.65i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.03 - 0.750i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.0591 - 0.182i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-3.94 + 2.86i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.05 + 1.49i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.05 - 3.25i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 7.07T + 23T^{2} \)
29 \( 1 + (-2.35 + 7.25i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.48 - 3.98i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.76 - 5.44i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.297 + 0.914i)T + (-33.1 + 24.0i)T^{2} \)
47 \( 1 + (0.0106 + 0.0326i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (9.24 - 6.71i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.69 + 8.30i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.47 + 1.06i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 5.26T + 67T^{2} \)
71 \( 1 + (4.21 + 3.06i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.428 + 1.31i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.23 - 0.894i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (9.94 + 7.22i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 1.80T + 89T^{2} \)
97 \( 1 + (-0.430 + 0.312i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.16506147285682137511499411327, −9.300536020577051011205226900639, −8.869785839903159596523201802850, −7.85976297558822023001797192995, −6.75955475771974708611461744834, −6.11776995591796468128996026966, −5.00710035538755115905160428501, −3.97383998026897144072644795252, −2.94963465916496861532095269857, −1.34589129351498520966895530711, 1.16193378424420012524628604360, 1.82765520954308399782896080978, 3.18829834781436322984129163361, 4.32618336128627435846288103419, 5.70160795793797612001916021745, 6.87052573296218535436078124071, 7.12269307955485083117433948152, 8.531105269074482113407430766511, 8.872554020684066204366735585685, 9.592063983763653460403856335837

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