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Properties

Label 2-294-147.104-c1-0-3
Degree $2$
Conductor $294$
Sign $0.0120 - 0.999i$
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 + (−1.69 − 0.367i)3-s + (0.222 + 0.974i)4-s + (2.13 + 1.02i)5-s + (−1.09 − 1.34i)6-s + (−1.53 + 2.15i)7-s + (−0.433 + 0.900i)8-s + (2.73 + 1.24i)9-s + (1.02 + 2.13i)10-s + (−2.95 − 2.35i)11-s + (−0.0184 − 1.73i)12-s + (3.59 + 2.86i)13-s + (−2.54 + 0.723i)14-s + (−3.23 − 2.52i)15-s + (−0.900 + 0.433i)16-s + (−0.623 + 2.73i)17-s + ⋯
L(s)  = 1  + (0.552 + 0.440i)2-s + (−0.977 − 0.212i)3-s + (0.111 + 0.487i)4-s + (0.955 + 0.460i)5-s + (−0.446 − 0.548i)6-s + (−0.581 + 0.813i)7-s + (−0.153 + 0.318i)8-s + (0.910 + 0.414i)9-s + (0.325 + 0.675i)10-s + (−0.891 − 0.710i)11-s + (−0.00533 − 0.499i)12-s + (0.997 + 0.795i)13-s + (−0.680 + 0.193i)14-s + (−0.836 − 0.652i)15-s + (−0.225 + 0.108i)16-s + (−0.151 + 0.663i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0120 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0120 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.0120 - 0.999i$
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: \((2,\ 294,\ (\ :1/2),\ 0.0120 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.963169 + 0.951606i\)
\(L(\frac12)\) \(\approx\) \(0.963169 + 0.951606i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 + 0.367i)T \)
7 \( 1 + (1.53 - 2.15i)T \)
good5 \( 1 + (-2.13 - 1.02i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (2.95 + 2.35i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-3.59 - 2.86i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (0.623 - 2.73i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 - 7.99iT - 19T^{2} \)
23 \( 1 + (-5.35 + 1.22i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (0.336 + 0.0767i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + 4.40iT - 31T^{2} \)
37 \( 1 + (-2.03 + 8.90i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (3.26 + 1.57i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-6.98 + 3.36i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-2.12 + 2.66i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (4.76 - 1.08i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (-9.34 + 4.50i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (7.60 + 1.73i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 - 6.99T + 67T^{2} \)
71 \( 1 + (11.0 - 2.52i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-7.61 + 6.07i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 4.27T + 79T^{2} \)
83 \( 1 + (-6.09 - 7.64i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (1.65 + 2.07i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 9.98iT - 97T^{2} \)
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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

−12.19583036803863313555301010665, −11.07630388428086603976629068558, −10.38886917737832975893803543130, −9.203844219211577983152279138839, −7.992214346274961872528130684798, −6.61512203907890132518893080380, −5.95756119565809933881233070943, −5.49207989909209481679266721049, −3.78951028744656125549643498125, −2.15007501817992297807555814739, 1.01907103103100452332646510066, 2.99623564368177113522731578050, 4.63349913303060675548405406123, 5.27015237173042891327661160529, 6.37586337703767651470092690295, 7.27414100497666506192654480282, 9.144574077495616476179018935170, 9.952061834584627070943990972442, 10.69368157545800421611174529456, 11.39233916350187614439668165215

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