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Properties

Label 2-1254-209.65-c1-0-33
Degree $2$
Conductor $1254$
Sign $-0.951 + 0.308i$
Analytic cond. $10.0132$
Root an. cond. $3.16437$
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  + (0.5 − 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (−2.18 + 3.78i)5-s + (0.866 − 0.499i)6-s − 2.88i·7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (2.18 + 3.78i)10-s + (−3.19 − 0.900i)11-s − 0.999i·12-s + (−1.29 − 2.24i)13-s + (−2.50 − 1.44i)14-s + (−3.78 + 2.18i)15-s + (−0.5 + 0.866i)16-s + (1.84 + 1.06i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.976 + 1.69i)5-s + (0.353 − 0.204i)6-s − 1.09i·7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.690 + 1.19i)10-s + (−0.962 − 0.271i)11-s − 0.288i·12-s + (−0.360 − 0.623i)13-s + (−0.668 − 0.385i)14-s + (−0.976 + 0.563i)15-s + (−0.125 + 0.216i)16-s + (0.446 + 0.257i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1254\)    =    \(2 \cdot 3 \cdot 11 \cdot 19\)
Sign: $-0.951 + 0.308i$
Analytic conductor: \(10.0132\)
Root analytic conductor: \(3.16437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1254} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1254,\ (\ :1/2),\ -0.951 + 0.308i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4980398113\)
\(L(\frac12)\) \(\approx\) \(0.4980398113\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (3.19 + 0.900i)T \)
19 \( 1 + (3.70 + 2.30i)T \)
good5 \( 1 + (2.18 - 3.78i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 2.88iT - 7T^{2} \)
13 \( 1 + (1.29 + 2.24i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.84 - 1.06i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.08 - 1.88i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.49 + 4.31i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.35iT - 31T^{2} \)
37 \( 1 - 4.74iT - 37T^{2} \)
41 \( 1 + (1.15 - 1.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.67 + 3.85i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.55 + 2.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.98 + 1.14i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.16 + 4.13i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.9 - 6.89i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.45 + 1.99i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.16 + 1.24i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-9.33 - 5.39i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.628 + 1.08i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.2iT - 83T^{2} \)
89 \( 1 + (13.5 - 7.79i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.65 + 3.84i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.824292863369524801159207973577, −8.273092653428186454729051981063, −7.70737728801213488140979807628, −7.03835163204500312507573996736, −6.00772113660564616107833125905, −4.66354120319327843055576217185, −3.80054867169156213284441106482, −3.17673399058683680884832321862, −2.36349158890872459047419404619, −0.16580089072450417861805258145, 1.72805871194295412115013906205, 3.10566753564374201734807486143, 4.27387160024016932820275145167, 5.00538697090743777708837157973, 5.63307493487385953862855546474, 6.93082270499200590087943181883, 7.80983426583572932651879124357, 8.383673198080292728090432666968, 8.915560470205306012548383323898, 9.588063753779319510162988596991

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