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Properties

Label 2-294-21.20-c1-0-6
Degree $2$
Conductor $294$
Sign $0.654 - 0.755i$
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  + i·2-s + 1.73·3-s − 4-s + 1.73·5-s + 1.73i·6-s − i·8-s + 2.99·9-s + 1.73i·10-s − 3i·11-s − 1.73·12-s + 3.46i·13-s + 2.99·15-s + 16-s − 3.46·17-s + 2.99i·18-s + 3.46i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.00·3-s − 0.5·4-s + 0.774·5-s + 0.707i·6-s − 0.353i·8-s + 0.999·9-s + 0.547i·10-s − 0.904i·11-s − 0.500·12-s + 0.960i·13-s + 0.774·15-s + 0.250·16-s − 0.840·17-s + 0.707i·18-s + 0.794i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.654 - 0.755i$
Analytic conductor: \(2.34760\)
Root analytic conductor: \(1.53218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ 0.654 - 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68229 + 0.768555i\)
\(L(\frac12)\) \(\approx\) \(1.68229 + 0.768555i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 - 1.73T \)
7 \( 1 \)
good5 \( 1 - 1.73T + 5T^{2} \)
11 \( 1 + 3iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 3iT - 29T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 6.92T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 + 9iT - 53T^{2} \)
59 \( 1 + 1.73T + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 + T + 79T^{2} \)
83 \( 1 - 8.66T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 - 5.19iT - 97T^{2} \)
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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

−12.12637958303247185410914475778, −10.67966599588227362581844130318, −9.751536703242806377712876485331, −8.843899690243860331944345809727, −8.290124679064220160569802673887, −6.96196056965239153782545959960, −6.19320793178905338163711428564, −4.80161848870963211906333674984, −3.54611384291359096692194646333, −1.96357637636041482382858833025, 1.78794214396375187233341014663, 2.83521829242009390007631835769, 4.16976536791737019015312599072, 5.39235071669289290881198338353, 6.93009175608147494389089303477, 7.999863597799318631084794763300, 9.075735472421335766308946612996, 9.765155804726530323009980557514, 10.44892148819749162360797691965, 11.67641328428386072586210348977

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