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Properties

Label 2-294-21.20-c5-0-15
Degree $2$
Conductor $294$
Sign $0.999 - 0.0324i$
Analytic cond. $47.1528$
Root an. cond. $6.86679$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4i·2-s + (9.81 − 12.1i)3-s − 16·4-s − 51.7·5-s + (−48.4 − 39.2i)6-s + 64i·8-s + (−50.2 − 237. i)9-s + 207. i·10-s + 453. i·11-s + (−157. + 193. i)12-s − 645. i·13-s + (−508. + 627. i)15-s + 256·16-s − 834.·17-s + (−951. + 200. i)18-s + 2.06e3i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.629 − 0.776i)3-s − 0.5·4-s − 0.926·5-s + (−0.549 − 0.445i)6-s + 0.353i·8-s + (−0.206 − 0.978i)9-s + 0.655i·10-s + 1.12i·11-s + (−0.314 + 0.388i)12-s − 1.05i·13-s + (−0.583 + 0.719i)15-s + 0.250·16-s − 0.700·17-s + (−0.691 + 0.146i)18-s + 1.31i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0324i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.999 - 0.0324i$
Analytic conductor: \(47.1528\)
Root analytic conductor: \(6.86679\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :5/2),\ 0.999 - 0.0324i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.201948940\)
\(L(\frac12)\) \(\approx\) \(1.201948940\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4iT \)
3 \( 1 + (-9.81 + 12.1i)T \)
7 \( 1 \)
good5 \( 1 + 51.7T + 3.12e3T^{2} \)
11 \( 1 - 453. iT - 1.61e5T^{2} \)
13 \( 1 + 645. iT - 3.71e5T^{2} \)
17 \( 1 + 834.T + 1.41e6T^{2} \)
19 \( 1 - 2.06e3iT - 2.47e6T^{2} \)
23 \( 1 - 754. iT - 6.43e6T^{2} \)
29 \( 1 - 1.45e3iT - 2.05e7T^{2} \)
31 \( 1 - 3.98e3iT - 2.86e7T^{2} \)
37 \( 1 - 1.54e4T + 6.93e7T^{2} \)
41 \( 1 - 1.94e4T + 1.15e8T^{2} \)
43 \( 1 + 6.77e3T + 1.47e8T^{2} \)
47 \( 1 - 5.36e3T + 2.29e8T^{2} \)
53 \( 1 - 3.48e4iT - 4.18e8T^{2} \)
59 \( 1 - 4.90e3T + 7.14e8T^{2} \)
61 \( 1 + 3.35e4iT - 8.44e8T^{2} \)
67 \( 1 + 2.75e4T + 1.35e9T^{2} \)
71 \( 1 - 4.48e4iT - 1.80e9T^{2} \)
73 \( 1 + 8.18e4iT - 2.07e9T^{2} \)
79 \( 1 + 3.38e4T + 3.07e9T^{2} \)
83 \( 1 + 9.45e3T + 3.93e9T^{2} \)
89 \( 1 - 8.84e4T + 5.58e9T^{2} \)
97 \( 1 - 1.01e4iT - 8.58e9T^{2} \)
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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

−11.07260983831163341659293095797, −10.00133690282660881346729121523, −9.030812774868939624924313573750, −7.907297846466102826507512179368, −7.49941914689337197729481507814, −6.03835561018479667686639057289, −4.44849930534302336033929774445, −3.45547828354290696308910902833, −2.31001110231419502015271748280, −1.04138169853802229399617551342, 0.35148171730217144320776804398, 2.63146794751460230312128931169, 3.97492914365580636663025835325, 4.56455530147846279227205031426, 5.98067185922619163417639304797, 7.20181048834147895204253597277, 8.164903651011575075574174809543, 8.878462643751553107915151536733, 9.649268718204002238974107128207, 11.10639919173603267654414838744

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