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Properties

Label 2-294-147.104-c1-0-10
Degree $2$
Conductor $294$
Sign $0.243 + 0.969i$
Analytic cond. $2.34760$
Root an. cond. $1.53218$
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  + (−0.781 − 0.623i)2-s + (−0.580 + 1.63i)3-s + (0.222 + 0.974i)4-s + (−3.14 − 1.51i)5-s + (1.47 − 0.913i)6-s + (2.47 − 0.926i)7-s + (0.433 − 0.900i)8-s + (−2.32 − 1.89i)9-s + (1.51 + 3.14i)10-s + (1.47 + 1.17i)11-s + (−1.72 − 0.202i)12-s + (−0.731 − 0.583i)13-s + (−2.51 − 0.820i)14-s + (4.30 − 4.25i)15-s + (−0.900 + 0.433i)16-s + (1.24 − 5.43i)17-s + ⋯
L(s)  = 1  + (−0.552 − 0.440i)2-s + (−0.335 + 0.942i)3-s + (0.111 + 0.487i)4-s + (−1.40 − 0.678i)5-s + (0.600 − 0.373i)6-s + (0.936 − 0.350i)7-s + (0.153 − 0.318i)8-s + (−0.775 − 0.631i)9-s + (0.479 + 0.996i)10-s + (0.445 + 0.355i)11-s + (−0.496 − 0.0585i)12-s + (−0.202 − 0.161i)13-s + (−0.672 − 0.219i)14-s + (1.11 − 1.09i)15-s + (−0.225 + 0.108i)16-s + (0.301 − 1.31i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.243 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.243 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.243 + 0.969i$
Analytic conductor: \(2.34760\)
Root analytic conductor: \(1.53218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ 0.243 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.498257 - 0.388685i\)
\(L(\frac12)\) \(\approx\) \(0.498257 - 0.388685i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.781 + 0.623i)T \)
3 \( 1 + (0.580 - 1.63i)T \)
7 \( 1 + (-2.47 + 0.926i)T \)
good5 \( 1 + (3.14 + 1.51i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-1.47 - 1.17i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.731 + 0.583i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-1.24 + 5.43i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 6.71iT - 19T^{2} \)
23 \( 1 + (-5.84 + 1.33i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-0.519 - 0.118i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + 5.34iT - 31T^{2} \)
37 \( 1 + (-0.612 + 2.68i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (1.61 + 0.777i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (9.41 - 4.53i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (7.08 - 8.88i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (5.46 - 1.24i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (-0.981 + 0.472i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-4.27 - 0.975i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 9.15T + 67T^{2} \)
71 \( 1 + (-5.70 + 1.30i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-1.01 + 0.808i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 15.3T + 79T^{2} \)
83 \( 1 + (9.92 + 12.4i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (6.59 + 8.26i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 7.02iT - 97T^{2} \)
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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

−11.44947925990528168429847592386, −10.93242132426393762104400484456, −9.598919014796610654428323728251, −8.892247687688822880825720066494, −7.939643819978705402786835378509, −7.01579240093803431488456989334, −4.87730489534937937848323494401, −4.52602556208582595280854484064, −3.15639485500433124885669418187, −0.63894086524695306657206939858, 1.56383304850974682905679116151, 3.50769482688225769727492445708, 5.15278803020394189002166341833, 6.38886125704584341150272421176, 7.23662883521428122109256628088, 8.180306922487351239671132643403, 8.474585320008620934760324513639, 10.33905364253900924564896339794, 11.21137825767593250102309621648, 11.78484288559739730392795366238

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