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Properties

Label 2-294-7.4-c5-0-2
Degree $2$
Conductor $294$
Sign $0.386 - 0.922i$
Analytic cond. $47.1528$
Root an. cond. $6.86679$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 3.46i)2-s + (−4.5 + 7.79i)3-s + (−7.99 + 13.8i)4-s + (−37.7 − 65.3i)5-s + 36·6-s + 63.9·8-s + (−40.5 − 70.1i)9-s + (−150. + 261. i)10-s + (74.7 − 129. i)11-s + (−72 − 124. i)12-s − 349.·13-s + 679.·15-s + (−128 − 221. i)16-s + (−574. + 995. i)17-s + (−162 + 280. i)18-s + (−1.39e3 − 2.42e3i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.675 − 1.16i)5-s + 0.408·6-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.477 + 0.826i)10-s + (0.186 − 0.322i)11-s + (−0.144 − 0.249i)12-s − 0.573·13-s + 0.779·15-s + (−0.125 − 0.216i)16-s + (−0.482 + 0.835i)17-s + (−0.117 + 0.204i)18-s + (−0.888 − 1.53i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(47.1528\)
Root analytic conductor: \(6.86679\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :5/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.2140345495\)
\(L(\frac12)\) \(\approx\) \(0.2140345495\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 + 3.46i)T \)
3 \( 1 + (4.5 - 7.79i)T \)
7 \( 1 \)
good5 \( 1 + (37.7 + 65.3i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-74.7 + 129. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 349.T + 3.71e5T^{2} \)
17 \( 1 + (574. - 995. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (1.39e3 + 2.42e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (906. + 1.57e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 759.T + 2.05e7T^{2} \)
31 \( 1 + (-4.51e3 + 7.82e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (3.89e3 + 6.75e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 7.64e3T + 1.15e8T^{2} \)
43 \( 1 - 1.21e4T + 1.47e8T^{2} \)
47 \( 1 + (-1.22e4 - 2.13e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (6.79e3 - 1.17e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.31e4 - 2.28e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.76e4 - 3.05e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.71e4 - 4.70e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 7.01e4T + 1.80e9T^{2} \)
73 \( 1 + (2.22e4 - 3.85e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (3.08e4 + 5.33e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 8.71e4T + 3.93e9T^{2} \)
89 \( 1 + (-4.92e4 - 8.53e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 3.23e4T + 8.58e9T^{2} \)
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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

−11.12040554232925858169306432081, −10.27217050949991608645598266359, −9.037355351624622303046758293008, −8.677560078546156721799952053921, −7.50662608048168609205794364016, −6.02060327468357356532514031778, −4.56845285496617647628659917294, −4.17146902968841297812886795462, −2.49557265211285296202468567236, −0.823118429253004335813725830263, 0.092742387068855774462298142356, 1.88723688085889009487778301271, 3.38478337584057942038021712558, 4.80177503316309585991159397724, 6.16221680042365728170294524695, 6.96270304934182491780245541264, 7.60265475787005885523531859905, 8.580969187360785353630292006044, 9.947377369977773761253544671824, 10.65580618653895184169797090806

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