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Properties

Label 2-294-147.104-c1-0-17
Degree $2$
Conductor $294$
Sign $-0.833 + 0.551i$
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 + (0.169 − 1.72i)3-s + (0.222 + 0.974i)4-s + (−0.210 − 0.101i)5-s + (−1.20 + 1.24i)6-s + (0.981 − 2.45i)7-s + (0.433 − 0.900i)8-s + (−2.94 − 0.585i)9-s + (0.101 + 0.210i)10-s + (−0.581 − 0.463i)11-s + (1.71 − 0.217i)12-s + (−0.948 − 0.756i)13-s + (−2.29 + 1.30i)14-s + (−0.210 + 0.345i)15-s + (−0.900 + 0.433i)16-s + (1.52 − 6.66i)17-s + ⋯
L(s)  = 1  + (−0.552 − 0.440i)2-s + (0.0981 − 0.995i)3-s + (0.111 + 0.487i)4-s + (−0.0941 − 0.0453i)5-s + (−0.492 + 0.506i)6-s + (0.370 − 0.928i)7-s + (0.153 − 0.318i)8-s + (−0.980 − 0.195i)9-s + (0.0320 + 0.0665i)10-s + (−0.175 − 0.139i)11-s + (0.496 − 0.0628i)12-s + (−0.263 − 0.209i)13-s + (−0.614 + 0.349i)14-s + (−0.0543 + 0.0892i)15-s + (−0.225 + 0.108i)16-s + (0.368 − 1.61i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 + 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.833 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.833 + 0.551i$
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: \((2,\ 294,\ (\ :1/2),\ -0.833 + 0.551i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.250731 - 0.833323i\)
\(L(\frac12)\) \(\approx\) \(0.250731 - 0.833323i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.169 + 1.72i)T \)
7 \( 1 + (-0.981 + 2.45i)T \)
good5 \( 1 + (0.210 + 0.101i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (0.581 + 0.463i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.948 + 0.756i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-1.52 + 6.66i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 - 4.63iT - 19T^{2} \)
23 \( 1 + (1.86 - 0.425i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (2.44 + 0.558i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 - 2.55iT - 31T^{2} \)
37 \( 1 + (-0.0943 + 0.413i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-5.27 - 2.53i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-1.04 + 0.503i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-3.40 + 4.27i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-11.3 + 2.58i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (-9.60 + 4.62i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-10.0 - 2.29i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 - 5.04T + 67T^{2} \)
71 \( 1 + (9.18 - 2.09i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-3.56 + 2.84i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 4.07T + 79T^{2} \)
83 \( 1 + (-5.46 - 6.84i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-8.77 - 11.0i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 3.82iT - 97T^{2} \)
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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

−11.55955776808879870787382300544, −10.47664836540197044038189127352, −9.575989177729960853917482569290, −8.297686241784791788467632280849, −7.65490272322301028526785747846, −6.86551931925107065660340386774, −5.42232864540425555047066821761, −3.77897747392821339076267087992, −2.33756519545755709169339416996, −0.77074970563555640723009917888, 2.32317658700830575029414916083, 3.99878719764406424960091107603, 5.27314210557849913886207289119, 6.05502121765452822663880445687, 7.57883472473350554672689182349, 8.549079242716617866820798970033, 9.218981057971385832142000188043, 10.14706308266924079027218838685, 11.04316572936864277861778542912, 11.84160374809899726884813405321

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