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Properties

Label 2-294-7.4-c5-0-14
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 + (27 + 46.7i)5-s + 36·6-s − 63.9·8-s + (−40.5 − 70.1i)9-s + (−108 + 187. i)10-s + (−108 + 187. i)11-s + (72 + 124. i)12-s + 998·13-s + 486·15-s + (−128 − 221. i)16-s + (−651 + 1.12e3i)17-s + (162 − 280. i)18-s + (−442 − 765. i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.482 + 0.836i)5-s + 0.408·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.341 + 0.591i)10-s + (−0.269 + 0.466i)11-s + (0.144 + 0.249i)12-s + 1.63·13-s + 0.557·15-s + (−0.125 − 0.216i)16-s + (−0.546 + 0.946i)17-s + (0.117 − 0.204i)18-s + (−0.280 − 0.486i)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

\(L(3)\) \(\approx\) \(2.534924110\)
\(L(\frac12)\) \(\approx\) \(2.534924110\)
\(L(\frac{7}{2})\) 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 \( 1 + (-27 - 46.7i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (108 - 187. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 998T + 3.71e5T^{2} \)
17 \( 1 + (651 - 1.12e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (442 + 765. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-1.13e3 - 1.96e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 1.48e3T + 2.05e7T^{2} \)
31 \( 1 + (4.18e3 - 7.23e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-2.35e3 - 4.08e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 9.78e3T + 1.15e8T^{2} \)
43 \( 1 - 1.94e4T + 1.47e8T^{2} \)
47 \( 1 + (1.11e4 + 1.92e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (1.33e4 - 2.32e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.40e4 - 2.43e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.94e4 - 3.36e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.19e4 - 2.07e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 2.06e4T + 1.80e9T^{2} \)
73 \( 1 + (145 - 251. i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-4.97e4 - 8.62e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 1.93e4T + 3.93e9T^{2} \)
89 \( 1 + (1.81e4 + 3.15e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 7.90e4T + 8.58e9T^{2} \)
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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

−11.17712706293237382766537884692, −10.45887107207067981584825193373, −9.086986168092864346785400502169, −8.318955084854022451046612049041, −7.13167683553929894505578171864, −6.47601181357573177289999655947, −5.56895972083949796888448728596, −4.01337010848736189811143009692, −2.89067461929700961009475227067, −1.53132886602871700989228006059, 0.57657508620634342274860678255, 1.89096844518616541336303696200, 3.26540448826440233691957047575, 4.34013109654423931777922691208, 5.35918646702774859097254865919, 6.28863286125263926998960559170, 8.053958179005049953078064103361, 8.995551550810860137681579070708, 9.530602274619590207947066410621, 10.85363310507144890709078205602

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