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Properties

Label 2-294-7.2-c5-0-5
Degree $2$
Conductor $294$
Sign $-0.900 + 0.435i$
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 + (−23.4 + 40.6i)5-s − 36·6-s + 63.9·8-s + (−40.5 + 70.1i)9-s + (−93.8 − 162. i)10-s + (43.7 + 75.7i)11-s + (72 − 124. i)12-s + 754.·13-s − 422.·15-s + (−128 + 221. i)16-s + (724. + 1.25e3i)17-s + (−162 − 280. i)18-s + (−1.27e3 + 2.19e3i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.419 + 0.727i)5-s − 0.408·6-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.296 − 0.514i)10-s + (0.108 + 0.188i)11-s + (0.144 − 0.249i)12-s + 1.23·13-s − 0.484·15-s + (−0.125 + 0.216i)16-s + (0.608 + 1.05i)17-s + (−0.117 − 0.204i)18-s + (−0.807 + 1.39i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.900 + 0.435i$
Analytic conductor: \(47.1528\)
Root analytic conductor: \(6.86679\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :5/2),\ -0.900 + 0.435i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.176155274\)
\(L(\frac12)\) \(\approx\) \(1.176155274\)
\(L(\frac{7}{2})\) 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 \( 1 + (23.4 - 40.6i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-43.7 - 75.7i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 754.T + 3.71e5T^{2} \)
17 \( 1 + (-724. - 1.25e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (1.27e3 - 2.19e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-456. + 790. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 173.T + 2.05e7T^{2} \)
31 \( 1 + (2.26e3 + 3.92e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (3.41e3 - 5.91e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 1.30e4T + 1.15e8T^{2} \)
43 \( 1 + 1.22e4T + 1.47e8T^{2} \)
47 \( 1 + (6.74e3 - 1.16e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-4.83e3 - 8.38e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (1.51e4 + 2.63e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (366. - 634. i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (2.31e4 + 4.01e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 3.29e3T + 1.80e9T^{2} \)
73 \( 1 + (5.11e3 + 8.86e3i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (4.92e4 - 8.53e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 8.77e4T + 3.93e9T^{2} \)
89 \( 1 + (-3.45e4 + 5.98e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 4.21e4T + 8.58e9T^{2} \)
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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

−11.04115310106611554450762195435, −10.53973989868688430335925150393, −9.544882874828582922254853343161, −8.414651356523668557045321119465, −7.84614529250371283784733395733, −6.56056129932081151486680461647, −5.76308750418008092983988042824, −4.20965514256745302725831824676, −3.35204625488756526396846502376, −1.56522024529313494043461495039, 0.37562797687600049170435455348, 1.30819071922384133055059544166, 2.76920660568826199405912575345, 3.93860864431990098658573006785, 5.16724962866889986672325826905, 6.64420643492239765187532363590, 7.71361956170166360376330762218, 8.741863026160127868231019191561, 9.104546410430042097618282233835, 10.52569008632735801476241808497

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