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Properties

Label 2-294-7.4-c5-0-8
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 + (43 + 74.4i)5-s − 36·6-s − 63.9·8-s + (−40.5 − 70.1i)9-s + (−172 + 297. i)10-s + (−17 + 29.4i)11-s + (−72 − 124. i)12-s + 3·13-s − 774.·15-s + (−128 − 221. i)16-s + (−952 + 1.64e3i)17-s + (162 − 280. i)18-s + (−744.5 − 1.28e3i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.769 + 1.33i)5-s − 0.408·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.543 + 0.942i)10-s + (−0.0423 + 0.0733i)11-s + (−0.144 − 0.249i)12-s + 0.00492·13-s − 0.888·15-s + (−0.125 − 0.216i)16-s + (−0.798 + 1.38i)17-s + (0.117 − 0.204i)18-s + (−0.473 − 0.819i)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

\(L(3)\) \(\approx\) \(1.169285261\)
\(L(\frac12)\) \(\approx\) \(1.169285261\)
\(L(\frac{7}{2})\) 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 \( 1 + (-43 - 74.4i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (17 - 29.4i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 3T + 3.71e5T^{2} \)
17 \( 1 + (952 - 1.64e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (744.5 + 1.28e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-112 - 193. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 6.50e3T + 2.05e7T^{2} \)
31 \( 1 + (-865.5 + 1.49e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-3.81e3 - 6.61e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.54e4T + 1.15e8T^{2} \)
43 \( 1 - 1.84e4T + 1.47e8T^{2} \)
47 \( 1 + (-9.23e3 - 1.59e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-9.97e3 + 1.72e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.59e4 - 2.75e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (2.88e4 + 4.99e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-3.02e4 + 5.24e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 4.48e4T + 1.80e9T^{2} \)
73 \( 1 + (-1.04e4 + 1.80e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-1.52e4 - 2.64e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 1.10e5T + 3.93e9T^{2} \)
89 \( 1 + (2.94e4 + 5.10e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.19e5T + 8.58e9T^{2} \)
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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

−11.27387736229650752234088444444, −10.71931116464294747510896777831, −9.768045403887808591763810122435, −8.774284993239319606535949538733, −7.44498123599887760610706494109, −6.43624624542158109095244120233, −5.91022029342992991163260945346, −4.56765105067927125306035083518, −3.39851478785159545803861195973, −2.14641973567871489123447945075, 0.27647657844965387612131433877, 1.40329686489076269739106779520, 2.41966581545510576503914260785, 4.19065861651640633201959575238, 5.23193735000298371912898166415, 5.93230521568187760191337984999, 7.29495143853881468371535156304, 8.682455427172272337749185291269, 9.306327293045087413871770529446, 10.35552324586099047025922338445

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