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Properties

Label 2-946-11.3-c1-0-10
Degree $2$
Conductor $946$
Sign $0.397 - 0.917i$
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 + (0.936 + 2.88i)3-s + (0.309 − 0.951i)4-s + (0.714 + 0.519i)5-s + (2.45 + 1.78i)6-s + (0.972 − 2.99i)7-s + (−0.309 − 0.951i)8-s + (−5.00 + 3.63i)9-s + 0.883·10-s + (2.53 + 2.13i)11-s + 3.02·12-s + (−4.68 + 3.40i)13-s + (−0.972 − 2.99i)14-s + (−0.827 + 2.54i)15-s + (−0.809 − 0.587i)16-s + (0.927 + 0.673i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.540 + 1.66i)3-s + (0.154 − 0.475i)4-s + (0.319 + 0.232i)5-s + (1.00 + 0.727i)6-s + (0.367 − 1.13i)7-s + (−0.109 − 0.336i)8-s + (−1.66 + 1.21i)9-s + 0.279·10-s + (0.765 + 0.643i)11-s + 0.874·12-s + (−1.30 + 0.944i)13-s + (−0.259 − 0.799i)14-s + (−0.213 + 0.657i)15-s + (−0.202 − 0.146i)16-s + (0.224 + 0.163i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(946\)    =    \(2 \cdot 11 \cdot 43\)
Sign: $0.397 - 0.917i$
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: \((2,\ 946,\ (\ :1/2),\ 0.397 - 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.30637 + 1.51479i\)
\(L(\frac12)\) \(\approx\) \(2.30637 + 1.51479i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.53 - 2.13i)T \)
43 \( 1 - T \)
good3 \( 1 + (-0.936 - 2.88i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-0.714 - 0.519i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.972 + 2.99i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (4.68 - 3.40i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.927 - 0.673i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.52 - 7.76i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 7.70T + 23T^{2} \)
29 \( 1 + (0.642 - 1.97i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.05 + 0.766i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.929 + 2.85i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.601 - 1.85i)T + (-33.1 + 24.0i)T^{2} \)
47 \( 1 + (3.99 + 12.2i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.50 + 1.82i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.947 - 2.91i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (7.58 + 5.51i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 2.18T + 67T^{2} \)
71 \( 1 + (-0.946 - 0.687i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.88 + 5.78i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.66 + 1.93i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (10.7 + 7.80i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 6.59T + 89T^{2} \)
97 \( 1 + (-3.85 + 2.79i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.03663654099533143855950847697, −9.797005953638025697415135566489, −8.916313473317393046387330084280, −7.67313006490514888374996365488, −6.76957420646207068400965446770, −5.41765259967489068973355363517, −4.55472518315006202344109764849, −4.05721679057484041358144449881, −3.16554139039175848116892648976, −1.83102785728470694991636885822, 1.10067606178806616559863173579, 2.57214469091096180244636679482, 2.99013111206553405118017694135, 4.98326312333518269786522113697, 5.61220241974530122615087405168, 6.58111336481427146699000030168, 7.30579416438049347938693679496, 7.989779692964489299052640490409, 8.944484492087391645122843701029, 9.331707165100273684889826333935

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