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Properties

Label 2-294-7.4-c5-0-18
Degree $2$
Conductor $294$
Sign $0.605 + 0.795i$
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 + (33 + 57.1i)5-s − 36·6-s + 63.9·8-s + (−40.5 − 70.1i)9-s + (132 − 228. i)10-s + (30 − 51.9i)11-s + (72 + 124. i)12-s − 658·13-s + 594·15-s + (−128 − 221. i)16-s + (207 − 358. i)17-s + (−162 + 280. i)18-s + (−478 − 827. i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.590 + 1.02i)5-s − 0.408·6-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.417 − 0.722i)10-s + (0.0747 − 0.129i)11-s + (0.144 + 0.249i)12-s − 1.07·13-s + 0.681·15-s + (−0.125 − 0.216i)16-s + (0.173 − 0.300i)17-s + (−0.117 + 0.204i)18-s + (−0.303 − 0.526i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.605 + 0.795i$
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: \((2,\ 294,\ (\ :5/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.929008316\)
\(L(\frac12)\) \(\approx\) \(1.929008316\)
\(L(\frac{7}{2})\) 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 \( 1 + (-33 - 57.1i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-30 + 51.9i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 658T + 3.71e5T^{2} \)
17 \( 1 + (-207 + 358. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (478 + 827. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (300 + 519. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 5.57e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.79e3 + 3.11e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-4.22e3 - 7.32e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 1.91e4T + 1.15e8T^{2} \)
43 \( 1 - 1.33e4T + 1.47e8T^{2} \)
47 \( 1 + (-9.84e3 - 1.70e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-1.56e4 + 2.70e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.31e4 - 2.28e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.55e4 - 2.69e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-8.40e3 + 1.45e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 6.12e3T + 1.80e9T^{2} \)
73 \( 1 + (-1.27e4 + 2.21e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (3.72e4 + 6.44e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 6.46e3T + 3.93e9T^{2} \)
89 \( 1 + (-1.63e4 - 2.83e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.66e5T + 8.58e9T^{2} \)
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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

−10.70164319644372487192447447604, −9.931922611168136564484775835678, −9.087613752775398076559848473026, −7.88039616301310903751840366734, −7.01913305727998631326584564022, −6.04566427734473286658921363417, −4.46568161348973520931320141865, −2.84261415547512086719710531418, −2.36407344547398290466223044517, −0.76277420677127378551363501070, 0.890096801724721471513467442261, 2.33572313192927834727374480225, 4.17554912735542614155739267163, 5.10197456290328277513606892993, 5.98789510655876439136376803698, 7.35466287442912369450250878399, 8.345583650632216559569705467367, 9.182239136121496288372983356757, 9.829114770807773502981612412812, 10.70751959053974537890711761330

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