Properties

Label 2-294-21.5-c1-0-7
Degree $2$
Conductor $294$
Sign $0.444 - 0.895i$
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  + (0.866 + 0.5i)2-s + (0.866 + 1.5i)3-s + (0.499 + 0.866i)4-s + (0.866 − 1.5i)5-s + 1.73i·6-s + 0.999i·8-s + (−1.5 + 2.59i)9-s + (1.5 − 0.866i)10-s + (2.59 − 1.5i)11-s + (−0.866 + 1.49i)12-s + 3.46i·13-s + 3·15-s + (−0.5 + 0.866i)16-s + (−1.73 − 3i)17-s + (−2.59 + 1.5i)18-s + (−3 − 1.73i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.499 + 0.866i)3-s + (0.249 + 0.433i)4-s + (0.387 − 0.670i)5-s + 0.707i·6-s + 0.353i·8-s + (−0.5 + 0.866i)9-s + (0.474 − 0.273i)10-s + (0.783 − 0.452i)11-s + (−0.250 + 0.433i)12-s + 0.960i·13-s + 0.774·15-s + (−0.125 + 0.216i)16-s + (−0.420 − 0.727i)17-s + (−0.612 + 0.353i)18-s + (−0.688 − 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.444 - 0.895i$
Analytic conductor: \(2.34760\)
Root analytic conductor: \(1.53218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ 0.444 - 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82583 + 1.13275i\)
\(L(\frac12)\) \(\approx\) \(1.82583 + 1.13275i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.866 - 1.5i)T \)
7 \( 1 \)
good5 \( 1 + (-0.866 + 1.5i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + (1.73 + 3i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3iT - 29T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.92T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (-3.46 + 6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.79 + 4.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.866 + 1.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 + (6 - 3.46i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.66T + 83T^{2} \)
89 \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \)
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

−11.90308771942846325802792840024, −11.15526562100158812281691042001, −9.874742102459122653052144052645, −9.004334432635046322601934927107, −8.381155258170212290019669483899, −6.91568104845687697071114094126, −5.78878148030345982863838181468, −4.64672727204623447297349475668, −3.91855333372078654525208845638, −2.31818064027532387097558645916, 1.70928767573067378324751072299, 2.89930122969985824905293277113, 4.10103047734304976859920472193, 5.87823666986884176736188306329, 6.52584454738619365605876139162, 7.60261531279878153504458801597, 8.703582523954608846158042171592, 9.912732168238700614617408849975, 10.73910933710601726844976579845, 11.89007876428997961393979613884

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