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Properties

Label 2-946-11.3-c1-0-19
Degree $2$
Conductor $946$
Sign $0.992 + 0.124i$
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 + (0.359 + 1.10i)3-s + (0.309 − 0.951i)4-s + (−0.0195 − 0.0141i)5-s + (0.942 + 0.684i)6-s + (0.0257 − 0.0792i)7-s + (−0.309 − 0.951i)8-s + (1.32 − 0.965i)9-s − 0.0241·10-s + (−1.33 + 3.03i)11-s + 1.16·12-s + (4.36 − 3.17i)13-s + (−0.0257 − 0.0792i)14-s + (0.00868 − 0.0267i)15-s + (−0.809 − 0.587i)16-s + (4.26 + 3.09i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.207 + 0.639i)3-s + (0.154 − 0.475i)4-s + (−0.00873 − 0.00634i)5-s + (0.384 + 0.279i)6-s + (0.00973 − 0.0299i)7-s + (−0.109 − 0.336i)8-s + (0.443 − 0.321i)9-s − 0.00763·10-s + (−0.403 + 0.914i)11-s + 0.336·12-s + (1.21 − 0.880i)13-s + (−0.00688 − 0.0211i)14-s + (0.00224 − 0.00690i)15-s + (−0.202 − 0.146i)16-s + (1.03 + 0.751i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(946\)    =    \(2 \cdot 11 \cdot 43\)
Sign: $0.992 + 0.124i$
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: \((2,\ 946,\ (\ :1/2),\ 0.992 + 0.124i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.54359 - 0.159161i\)
\(L(\frac12)\) \(\approx\) \(2.54359 - 0.159161i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.33 - 3.03i)T \)
43 \( 1 - T \)
good3 \( 1 + (-0.359 - 1.10i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (0.0195 + 0.0141i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.0257 + 0.0792i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-4.36 + 3.17i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.26 - 3.09i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.99 - 6.14i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 7.91T + 23T^{2} \)
29 \( 1 + (-2.64 + 8.13i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-6.27 + 4.56i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.862 + 2.65i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.16 - 6.66i)T + (-33.1 + 24.0i)T^{2} \)
47 \( 1 + (-1.11 - 3.44i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.230 + 0.167i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.57 - 7.93i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.27 + 1.65i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 3.48T + 67T^{2} \)
71 \( 1 + (7.00 + 5.09i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.967 - 2.97i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (7.18 - 5.21i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (11.2 + 8.17i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 1.29T + 89T^{2} \)
97 \( 1 + (11.0 - 8.01i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.09332207776361647252986663733, −9.687510486186070990698483145127, −8.190929684007157849135033939794, −7.76898520100591906926427990624, −6.13979263479938492011819890011, −5.79556861258238643622761182632, −4.25543749739902613647947788982, −3.98779714229736283342229629733, −2.76437985389099937175221514104, −1.32935794284046465973404663281, 1.29176413615707148039512873248, 2.73627960814832866998682765525, 3.72752657198310721451182752027, 4.90255365575116446517177114605, 5.77135793048319419986808985313, 6.74886714208272624180009037951, 7.35583789171701590797541570646, 8.286536917820537605863916973402, 8.915003053016392324952203831401, 10.10533791141019651627260445211

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