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Properties

Label 2-946-473.7-c1-0-27
Degree $2$
Conductor $946$
Sign $-0.376 + 0.926i$
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 + (−0.599 − 2.81i)3-s + (0.309 + 0.951i)4-s + (0.127 − 0.286i)5-s + (−1.17 + 2.63i)6-s + (3.11 + 0.661i)7-s + (0.309 − 0.951i)8-s + (−4.85 + 2.16i)9-s + (−0.271 + 0.156i)10-s + (3.13 − 1.07i)11-s + (2.49 − 1.44i)12-s + (1.17 + 2.64i)13-s + (−2.12 − 2.36i)14-s + (−0.885 − 0.188i)15-s + (−0.809 + 0.587i)16-s + (7.42 − 0.780i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.346 − 1.62i)3-s + (0.154 + 0.475i)4-s + (0.0571 − 0.128i)5-s + (−0.478 + 1.07i)6-s + (1.17 + 0.249i)7-s + (0.109 − 0.336i)8-s + (−1.61 + 0.720i)9-s + (−0.0859 + 0.0496i)10-s + (0.945 − 0.325i)11-s + (0.720 − 0.416i)12-s + (0.326 + 0.732i)13-s + (−0.568 − 0.631i)14-s + (−0.228 − 0.0485i)15-s + (−0.202 + 0.146i)16-s + (1.80 − 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.376 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(946\)    =    \(2 \cdot 11 \cdot 43\)
Sign: $-0.376 + 0.926i$
Analytic conductor: \(7.55384\)
Root analytic conductor: \(2.74842\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{946} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 946,\ (\ :1/2),\ -0.376 + 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.755683 - 1.12305i\)
\(L(\frac12)\) \(\approx\) \(0.755683 - 1.12305i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.13 + 1.07i)T \)
43 \( 1 + (4.39 - 4.86i)T \)
good3 \( 1 + (0.599 + 2.81i)T + (-2.74 + 1.22i)T^{2} \)
5 \( 1 + (-0.127 + 0.286i)T + (-3.34 - 3.71i)T^{2} \)
7 \( 1 + (-3.11 - 0.661i)T + (6.39 + 2.84i)T^{2} \)
13 \( 1 + (-1.17 - 2.64i)T + (-8.69 + 9.66i)T^{2} \)
17 \( 1 + (-7.42 + 0.780i)T + (16.6 - 3.53i)T^{2} \)
19 \( 1 + (1.27 - 0.269i)T + (17.3 - 7.72i)T^{2} \)
23 \( 1 + (2.18 + 3.78i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-9.62 - 2.04i)T + (26.4 + 11.7i)T^{2} \)
31 \( 1 + (-0.635 + 6.04i)T + (-30.3 - 6.44i)T^{2} \)
37 \( 1 + (1.41 + 1.27i)T + (3.86 + 36.7i)T^{2} \)
41 \( 1 + (-1.74 - 0.567i)T + (33.1 + 24.0i)T^{2} \)
47 \( 1 + (0.926 - 2.85i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.691 - 6.57i)T + (-51.8 - 11.0i)T^{2} \)
59 \( 1 + (0.160 + 0.493i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.302 - 2.88i)T + (-59.6 + 12.6i)T^{2} \)
67 \( 1 + (3.33 + 5.78i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.316 + 0.0332i)T + (69.4 - 14.7i)T^{2} \)
73 \( 1 + (6.91 + 1.47i)T + (66.6 + 29.6i)T^{2} \)
79 \( 1 + (4.04 + 9.09i)T + (-52.8 + 58.7i)T^{2} \)
83 \( 1 + (4.86 - 10.9i)T + (-55.5 - 61.6i)T^{2} \)
89 \( 1 + (9.95 - 5.74i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.58 + 6.96i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.730319065645464316891560571284, −8.622708164754471260204461016776, −8.187023600811451787775994137941, −7.37302520871164476829806280602, −6.52567639234905128606213353957, −5.73065482251024855521488915023, −4.45970849292349260109882741159, −2.89999409765595740853059108992, −1.61721760122197668054088261510, −1.08622841466421164502877902242, 1.25784544022226291740626805535, 3.22228927651892556335897577917, 4.26647229650416998004806731377, 5.08233390425676111974190478346, 5.79181880449478103037005623456, 6.91307842975735793303632137537, 8.160422290247182470967012730908, 8.563551705507977762287827648551, 9.718011956535159771208356651448, 10.21888067032196802323369420960

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