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Properties

Label 2-294-7.2-c3-0-2
Degree $2$
Conductor $294$
Sign $-0.605 + 0.795i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(17.3465\)
Root analytic conductor: \(4.16492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :3/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8240961275\)
\(L(\frac12)\) \(\approx\) \(0.8240961275\)
\(L(\frac{5}{2})\) 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

−11.80256734386607916076953020427, −10.74400079082406665016746455483, −10.03485331494991991588894031302, −9.180785358798141723670426509418, −7.900240437266681571238482402224, −7.14537965561811192737058032971, −6.43198463216074979303590465020, −4.70535769515924583521808301096, −3.79334438625202002811973613769, −2.26879799971617326405731060334, 0.35841161437483477061788456994, 1.35989917876441667622530489683, 3.18039436649001069729413486080, 4.26155008768366600170803172520, 5.54665003287156597521252160844, 7.09887533828495161041919294615, 8.153946711848378336440166351558, 8.871008163904826320323086605753, 9.390962628134646597837617319495, 11.15180655272409703782697519361

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