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Properties

Label 2-294-7.4-c5-0-25
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 + (−3 − 5.19i)5-s − 36·6-s + 63.9·8-s + (−40.5 − 70.1i)9-s + (−12 + 20.7i)10-s + (333 − 576. i)11-s + (72 + 124. i)12-s + 559·13-s − 54·15-s + (−128 − 221. i)16-s + (−870 + 1.50e3i)17-s + (−162 + 280. i)18-s + (578.5 + 1.00e3i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.0536 − 0.0929i)5-s − 0.408·6-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.0379 + 0.0657i)10-s + (0.829 − 1.43i)11-s + (0.144 + 0.249i)12-s + 0.917·13-s − 0.0619·15-s + (−0.125 − 0.216i)16-s + (−0.730 + 1.26i)17-s + (−0.117 + 0.204i)18-s + (0.367 + 0.636i)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

\(L(3)\) \(\approx\) \(1.932179518\)
\(L(\frac12)\) \(\approx\) \(1.932179518\)
\(L(\frac{7}{2})\) 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 \( 1 + (3 + 5.19i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-333 + 576. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 559T + 3.71e5T^{2} \)
17 \( 1 + (870 - 1.50e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-578.5 - 1.00e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-1.73e3 - 3.00e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 3.37e3T + 2.05e7T^{2} \)
31 \( 1 + (-3.14e3 + 5.44e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (1.56e3 + 2.71e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 4.86e3T + 1.15e8T^{2} \)
43 \( 1 + 1.14e4T + 1.47e8T^{2} \)
47 \( 1 + (-1.15e3 - 2.00e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-1.41e4 + 2.45e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.02e4 + 1.77e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (2.31e3 + 4.00e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-9.37e3 + 1.62e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 3.82e4T + 1.80e9T^{2} \)
73 \( 1 + (-3.52e4 + 6.11e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-3.11e4 - 5.39e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 7.98e4T + 3.93e9T^{2} \)
89 \( 1 + (9.06e3 + 1.56e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.24e5T + 8.58e9T^{2} \)
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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

−10.85368577207167651578553297938, −9.621875835566487876493018268320, −8.539866121027255912603647154679, −8.236451994783072162007565001063, −6.70050611261365736480099378246, −5.82678148160099584910215572988, −4.02709157117137369075997123664, −3.18479114723959621418810255129, −1.66136784998079231227713008756, −0.69570274335452160640611942480, 1.11591644617717784554228576472, 2.75980179333151237766430056604, 4.31195164346514827120201283176, 5.06349750084131179402539717566, 6.66340240983807495978096858029, 7.16229498861651554853591421634, 8.616489108274838051287344050981, 9.155231460146695749733615012799, 10.08322346189933902031858603593, 11.03567748981070995046271770462

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