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Properties

Label 2-946-11.3-c1-0-20
Degree $2$
Conductor $946$
Sign $0.906 + 0.421i$
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 + (0.716 + 2.20i)3-s + (0.309 − 0.951i)4-s + (−2.37 − 1.72i)5-s + (1.87 + 1.36i)6-s + (−0.0358 + 0.110i)7-s + (−0.309 − 0.951i)8-s + (−1.92 + 1.39i)9-s − 2.93·10-s + (2.20 − 2.48i)11-s + 2.31·12-s + (1.90 − 1.38i)13-s + (0.0358 + 0.110i)14-s + (2.10 − 6.48i)15-s + (−0.809 − 0.587i)16-s + (5.75 + 4.18i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.413 + 1.27i)3-s + (0.154 − 0.475i)4-s + (−1.06 − 0.772i)5-s + (0.766 + 0.556i)6-s + (−0.0135 + 0.0416i)7-s + (−0.109 − 0.336i)8-s + (−0.642 + 0.466i)9-s − 0.929·10-s + (0.663 − 0.748i)11-s + 0.669·12-s + (0.527 − 0.382i)13-s + (0.00957 + 0.0294i)14-s + (0.543 − 1.67i)15-s + (−0.202 − 0.146i)16-s + (1.39 + 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(946\)    =    \(2 \cdot 11 \cdot 43\)
Sign: $0.906 + 0.421i$
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: \((2,\ 946,\ (\ :1/2),\ 0.906 + 0.421i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.19284 - 0.484869i\)
\(L(\frac12)\) \(\approx\) \(2.19284 - 0.484869i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.20 + 2.48i)T \)
43 \( 1 + T \)
good3 \( 1 + (-0.716 - 2.20i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (2.37 + 1.72i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.0358 - 0.110i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.90 + 1.38i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-5.75 - 4.18i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.19 + 3.68i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 4.60T + 23T^{2} \)
29 \( 1 + (-0.507 + 1.56i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.65 + 2.65i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.31 - 4.05i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.73 + 5.33i)T + (-33.1 + 24.0i)T^{2} \)
47 \( 1 + (2.22 + 6.83i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.572 + 0.416i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.37 - 10.3i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.15 + 3.74i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 7.69T + 67T^{2} \)
71 \( 1 + (5.68 + 4.13i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.73 + 8.41i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.68 - 4.13i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-9.85 - 7.15i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 3.01T + 89T^{2} \)
97 \( 1 + (-14.0 + 10.2i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.18204858404977546200652745274, −9.029095666203253418228394433452, −8.658699799968555414960660290910, −7.70071381137662419713579580589, −6.28512226446696025397499992851, −5.25318373640968062168837326397, −4.41505342728221476892475917858, −3.72161510091207919955802824007, −3.10804508885981999824489197434, −1.01061930872527452511816710966, 1.39249805700996253300713180499, 2.86009477264841548498127249402, 3.63996258897415732160963644089, 4.75986776179184331342301877709, 6.13743719113087776142358874691, 6.93417252570354668717450027704, 7.40123669966490161659537081202, 7.990656952270794976906557227967, 8.947044926152013571687542201818, 10.15569454860204363661052447346

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