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Properties

Label 2-294-21.2-c2-0-9
Degree $2$
Conductor $294$
Sign $0.987 + 0.158i$
Analytic cond. $8.01091$
Root an. cond. $2.83035$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.707i)2-s + (2.54 + 1.58i)3-s + (0.999 + 1.73i)4-s + (−5.68 − 3.28i)5-s + (−1.99 − 3.74i)6-s − 2.82i·8-s + (3.98 + 8.07i)9-s + (4.64 + 8.04i)10-s + (0.357 − 0.206i)11-s + (−0.196 + 5.99i)12-s + 20.5·13-s + (−9.29 − 17.3i)15-s + (−2.00 + 3.46i)16-s + (13.7 − 7.93i)17-s + (0.832 − 12.7i)18-s + (8 − 13.8i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.849 + 0.528i)3-s + (0.249 + 0.433i)4-s + (−1.13 − 0.657i)5-s + (−0.333 − 0.623i)6-s − 0.353i·8-s + (0.442 + 0.896i)9-s + (0.464 + 0.804i)10-s + (0.0324 − 0.0187i)11-s + (−0.0163 + 0.499i)12-s + 1.58·13-s + (−0.619 − 1.15i)15-s + (−0.125 + 0.216i)16-s + (0.808 − 0.467i)17-s + (0.0462 − 0.705i)18-s + (0.421 − 0.729i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.987 + 0.158i$
Analytic conductor: \(8.01091\)
Root analytic conductor: \(2.83035\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1),\ 0.987 + 0.158i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.45083 - 0.115925i\)
\(L(\frac12)\) \(\approx\) \(1.45083 - 0.115925i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 + (-2.54 - 1.58i)T \)
7 \( 1 \)
good5 \( 1 + (5.68 + 3.28i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-0.357 + 0.206i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 20.5T + 169T^{2} \)
17 \( 1 + (-13.7 + 7.93i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-8 + 13.8i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-31.1 - 18.0i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 20.8iT - 841T^{2} \)
31 \( 1 + (-2.77 - 4.79i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (10 - 17.3i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 76.1iT - 1.68e3T^{2} \)
43 \( 1 - 51.7T + 1.84e3T^{2} \)
47 \( 1 + (-7.34 - 4.24i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (44.0 - 25.4i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (1.42 - 0.824i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-33.4 + 57.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (24.7 + 42.7i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 87.7iT - 5.04e3T^{2} \)
73 \( 1 + (-6.16 - 10.6i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-42.4 + 73.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 4.12iT - 6.88e3T^{2} \)
89 \( 1 + (27.0 + 15.6i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 68.8T + 9.40e3T^{2} \)
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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

−11.29699108913895061396126731242, −10.67655879127716309526401526890, −9.319619255301785830873277306945, −8.814547347525931158950878343251, −7.973770176920776602777964210624, −7.14088967575965145511043355351, −5.18755010336256427673913783083, −3.92181221532003982712100595502, −3.12352061763795136720779507763, −1.12165303810224317291297010059, 1.14885814929620886645444178517, 3.02349357292368682893727408174, 3.99091994226387915465234989758, 6.03426748886586270348351718670, 6.97530235793747774548712379573, 7.904702025624492626974736864664, 8.383708692858983340989952137169, 9.470474585267255961254640143510, 10.65184753187617256026705246970, 11.44289827943726863623081372007

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