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Properties

Label 2-294-147.41-c1-0-1
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: $\chi_{294} (41, \cdot )$
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

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

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