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Properties

Label 2-946-11.3-c1-0-18
Degree $2$
Conductor $946$
Sign $-0.415 + 0.909i$
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 + (−1.04 − 3.21i)3-s + (0.309 − 0.951i)4-s + (1.50 + 1.09i)5-s + (2.73 + 1.98i)6-s + (−1.11 + 3.43i)7-s + (0.309 + 0.951i)8-s + (−6.82 + 4.95i)9-s − 1.86·10-s + (2.49 + 2.18i)11-s − 3.38·12-s + (2.15 − 1.56i)13-s + (−1.11 − 3.43i)14-s + (1.94 − 5.98i)15-s + (−0.809 − 0.587i)16-s + (−5.16 − 3.75i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.603 − 1.85i)3-s + (0.154 − 0.475i)4-s + (0.673 + 0.489i)5-s + (1.11 + 0.811i)6-s + (−0.422 + 1.29i)7-s + (0.109 + 0.336i)8-s + (−2.27 + 1.65i)9-s − 0.588·10-s + (0.752 + 0.658i)11-s − 0.976·12-s + (0.597 − 0.434i)13-s + (−0.298 − 0.918i)14-s + (0.502 − 1.54i)15-s + (−0.202 − 0.146i)16-s + (−1.25 − 0.910i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.415 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(946\)    =    \(2 \cdot 11 \cdot 43\)
Sign: $-0.415 + 0.909i$
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: \((2,\ 946,\ (\ :1/2),\ -0.415 + 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.397593 - 0.618588i\)
\(L(\frac12)\) \(\approx\) \(0.397593 - 0.618588i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.49 - 2.18i)T \)
43 \( 1 - T \)
good3 \( 1 + (1.04 + 3.21i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.50 - 1.09i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (1.11 - 3.43i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-2.15 + 1.56i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (5.16 + 3.75i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.14 + 6.58i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.23T + 23T^{2} \)
29 \( 1 + (-2.51 + 7.73i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.58 + 2.60i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.845 + 2.60i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.21 + 9.90i)T + (-33.1 + 24.0i)T^{2} \)
47 \( 1 + (0.100 + 0.308i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.18 - 3.04i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.86 + 5.75i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (7.69 + 5.58i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 13.8T + 67T^{2} \)
71 \( 1 + (-4.10 - 2.97i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.15 + 6.63i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (2.45 - 1.78i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-11.3 - 8.25i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 3.01T + 89T^{2} \)
97 \( 1 + (2.53 - 1.83i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.489925866150600303817837101067, −8.841201382919131653017805188326, −8.010256645454765501702956635485, −6.94073028576802286442215648757, −6.45109909075527437739103148705, −6.03599038415920267187843742858, −4.99016169843318957290339452918, −2.41033193737595920015171399888, −2.23206945055803381921937246816, −0.47745289026643334397185181329, 1.29041857416517740451102929851, 3.41943888907405304749616834172, 3.95400332661525512266832640706, 4.76861878215967923136818914944, 6.10350510508705162570006509369, 6.53022927738031450669177735556, 8.391406825773363330281611106328, 8.881568513173571478506280993739, 9.729800174008479844110309250803, 10.25728210041676781975533253575

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