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Properties

Label 2-294-7.4-c3-0-9
Degree $2$
Conductor $294$
Sign $0.701 - 0.712i$
Analytic cond. $17.3465$
Root an. cond. $4.16492$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (−1.5 + 2.59i)3-s + (−1.99 + 3.46i)4-s + (9 + 15.5i)5-s + 6·6-s + 7.99·8-s + (−4.5 − 7.79i)9-s + (18 − 31.1i)10-s + (36 − 62.3i)11-s + (−6.00 − 10.3i)12-s + 34·13-s − 54·15-s + (−8 − 13.8i)16-s + (3 − 5.19i)17-s + (−9 + 15.5i)18-s + (46 + 79.6i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.804 + 1.39i)5-s + 0.408·6-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.569 − 0.985i)10-s + (0.986 − 1.70i)11-s + (−0.144 − 0.249i)12-s + 0.725·13-s − 0.929·15-s + (−0.125 − 0.216i)16-s + (0.0428 − 0.0741i)17-s + (−0.117 + 0.204i)18-s + (0.555 + 0.962i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.701 - 0.712i$
Analytic conductor: \(17.3465\)
Root analytic conductor: \(4.16492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :3/2),\ 0.701 - 0.712i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.658992295\)
\(L(\frac12)\) \(\approx\) \(1.658992295\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
3 \( 1 + (1.5 - 2.59i)T \)
7 \( 1 \)
good5 \( 1 + (-9 - 15.5i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-36 + 62.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 34T + 2.19e3T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-46 - 79.6i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-90 - 155. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 114T + 2.43e4T^{2} \)
31 \( 1 + (-28 + 48.4i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-17 - 29.4i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 6T + 6.89e4T^{2} \)
43 \( 1 - 164T + 7.95e4T^{2} \)
47 \( 1 + (-84 - 145. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (327 - 566. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (246 - 426. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (125 + 216. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-62 + 107. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 36T + 3.57e5T^{2} \)
73 \( 1 + (-505 + 874. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (28 + 48.4i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 228T + 5.71e5T^{2} \)
89 \( 1 + (-195 - 337. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 70T + 9.12e5T^{2} \)
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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

−11.03734398755813892155195086034, −10.83367490101292130828732611108, −9.638122158146124879964186113467, −9.034207350841832071473549685552, −7.65169790269441900313709928299, −6.31025207360612070799083418085, −5.69879605315703122587235905429, −3.69870666270017911564391267134, −3.07643533580961143096863002898, −1.31280186971507764460976431504, 0.867351220233494668367279403138, 1.89657338992337791702348011604, 4.45747563539411154158550571885, 5.22711813677678615058511494478, 6.39427734182107692123224836449, 7.18421324607027571236106716280, 8.498272002996662019431318985818, 9.195666317092763118611955588336, 9.915611450390347093717415859951, 11.26292736542813558040699662233

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