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Properties

Label 2-946-11.3-c1-0-13
Degree $2$
Conductor $946$
Sign $0.991 - 0.132i$
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 + (−0.367 − 1.13i)3-s + (0.309 − 0.951i)4-s + (3.00 + 2.18i)5-s + (0.963 + 0.699i)6-s + (−0.576 + 1.77i)7-s + (0.309 + 0.951i)8-s + (1.28 − 0.930i)9-s − 3.72·10-s + (0.520 − 3.27i)11-s − 1.19·12-s + (2.34 − 1.70i)13-s + (−0.576 − 1.77i)14-s + (1.36 − 4.21i)15-s + (−0.809 − 0.587i)16-s + (0.172 + 0.125i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.212 − 0.653i)3-s + (0.154 − 0.475i)4-s + (1.34 + 0.978i)5-s + (0.393 + 0.285i)6-s + (−0.217 + 0.670i)7-s + (0.109 + 0.336i)8-s + (0.426 − 0.310i)9-s − 1.17·10-s + (0.156 − 0.987i)11-s − 0.343·12-s + (0.649 − 0.471i)13-s + (−0.153 − 0.473i)14-s + (0.353 − 1.08i)15-s + (−0.202 − 0.146i)16-s + (0.0418 + 0.0304i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(946\)    =    \(2 \cdot 11 \cdot 43\)
Sign: $0.991 - 0.132i$
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: \((2,\ 946,\ (\ :1/2),\ 0.991 - 0.132i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51524 + 0.100722i\)
\(L(\frac12)\) \(\approx\) \(1.51524 + 0.100722i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.520 + 3.27i)T \)
43 \( 1 - T \)
good3 \( 1 + (0.367 + 1.13i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-3.00 - 2.18i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.576 - 1.77i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-2.34 + 1.70i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.172 - 0.125i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.698 + 2.15i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 6.98T + 23T^{2} \)
29 \( 1 + (1.11 - 3.43i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (7.33 - 5.32i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.20 + 6.78i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.00918 + 0.0282i)T + (-33.1 + 24.0i)T^{2} \)
47 \( 1 + (-0.0181 - 0.0557i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.19 + 2.32i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.44 + 7.53i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-6.58 - 4.78i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 7.67T + 67T^{2} \)
71 \( 1 + (-6.07 - 4.41i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.92 - 12.0i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (3.19 - 2.32i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (9.92 + 7.20i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 1.66T + 89T^{2} \)
97 \( 1 + (-4.32 + 3.14i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.956851683372636322294914495016, −9.150295750488169365160515637428, −8.582354876311554161625212780282, −7.14765758741831609549078299514, −6.79565263119098813845094718603, −5.85615482261143685723196403797, −5.48465618509400412569067634963, −3.39985108147066854674555069590, −2.34303308560074990651109812296, −1.13305854768200823937828842253, 1.22871206819307235008320749907, 2.13735422741580103082480845014, 3.86363810546313560238758931892, 4.64596952192431467791068456181, 5.55186610080634588879001842403, 6.66334961959571136976362075419, 7.57603560493609265281038205851, 8.762660016748839616054473367338, 9.467700375873700120808400473857, 9.881188042201741185667210341593

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