Properties

Label 2-294-147.104-c1-0-18
Degree $2$
Conductor $294$
Sign $-0.998 - 0.0611i$
Analytic cond. $2.34760$
Root an. cond. $1.53218$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.781 − 0.623i)2-s + (1.06 − 1.36i)3-s + (0.222 + 0.974i)4-s + (−2.67 − 1.28i)5-s + (−1.68 + 0.405i)6-s + (−1.08 + 2.41i)7-s + (0.433 − 0.900i)8-s + (−0.736 − 2.90i)9-s + (1.28 + 2.67i)10-s + (−3.64 − 2.90i)11-s + (1.56 + 0.733i)12-s + (−2.21 − 1.76i)13-s + (2.35 − 1.21i)14-s + (−4.60 + 2.28i)15-s + (−0.900 + 0.433i)16-s + (−1.10 + 4.85i)17-s + ⋯
L(s)  = 1  + (−0.552 − 0.440i)2-s + (0.614 − 0.789i)3-s + (0.111 + 0.487i)4-s + (−1.19 − 0.576i)5-s + (−0.687 + 0.165i)6-s + (−0.408 + 0.912i)7-s + (0.153 − 0.318i)8-s + (−0.245 − 0.969i)9-s + (0.407 + 0.846i)10-s + (−1.10 − 0.877i)11-s + (0.453 + 0.211i)12-s + (−0.614 − 0.489i)13-s + (0.628 − 0.324i)14-s + (−1.18 + 0.590i)15-s + (−0.225 + 0.108i)16-s + (−0.268 + 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0611i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.998 - 0.0611i$
Analytic conductor: \(2.34760\)
Root analytic conductor: \(1.53218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ -0.998 - 0.0611i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0154445 + 0.504678i\)
\(L(\frac12)\) \(\approx\) \(0.0154445 + 0.504678i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.781 + 0.623i)T \)
3 \( 1 + (-1.06 + 1.36i)T \)
7 \( 1 + (1.08 - 2.41i)T \)
good5 \( 1 + (2.67 + 1.28i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (3.64 + 2.90i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (2.21 + 1.76i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (1.10 - 4.85i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 6.81iT - 19T^{2} \)
23 \( 1 + (-2.40 + 0.549i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-5.53 - 1.26i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 - 7.19iT - 31T^{2} \)
37 \( 1 + (-1.90 + 8.34i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (7.72 + 3.72i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (2.96 - 1.42i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-2.98 + 3.74i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-9.35 + 2.13i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (0.790 - 0.380i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (5.12 + 1.17i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 - 2.52T + 67T^{2} \)
71 \( 1 + (-7.21 + 1.64i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-13.1 + 10.4i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 0.959T + 79T^{2} \)
83 \( 1 + (0.996 + 1.25i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (4.90 + 6.14i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 2.75iT - 97T^{2} \)
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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

−11.43618148321449905607184564602, −10.43233614077938690756515549387, −8.911157566847356410704602538376, −8.573498234337362479411454282173, −7.77139522366871764342649993956, −6.69656462006776355860969128809, −5.15209932732823756518285000166, −3.45143640712972605823397284555, −2.50550916960528350000316353421, −0.39636045188320678837237013039, 2.72965152328900952365316853267, 4.02696219502499025295515476354, 4.97238216335329988130827234963, 6.84430443158774492718130291146, 7.62544148298467154658758331761, 8.170157702544844018988036590177, 9.700309424614532108304700958394, 10.07947815835647422058174128276, 11.03016043011116475577007060241, 12.00334711647037156905157691414

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