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Properties

Label 2-294-21.11-c2-0-20
Degree $2$
Conductor $294$
Sign $0.298 + 0.954i$
Analytic cond. $8.01091$
Root an. cond. $2.83035$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 0.707i)2-s + (2.54 − 1.58i)3-s + (0.999 − 1.73i)4-s + (0.790 − 0.456i)5-s + (1.99 − 3.74i)6-s − 2.82i·8-s + (3.98 − 8.07i)9-s + (0.645 − 1.11i)10-s + (12.6 + 7.27i)11-s + (−0.196 − 5.99i)12-s + 0.583·13-s + (1.29 − 2.41i)15-s + (−2.00 − 3.46i)16-s + (−18.6 − 10.7i)17-s + (−0.832 − 12.7i)18-s + (−8 − 13.8i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.849 − 0.528i)3-s + (0.249 − 0.433i)4-s + (0.158 − 0.0913i)5-s + (0.333 − 0.623i)6-s − 0.353i·8-s + (0.442 − 0.896i)9-s + (0.0645 − 0.111i)10-s + (1.14 + 0.661i)11-s + (−0.0163 − 0.499i)12-s + 0.0448·13-s + (0.0861 − 0.161i)15-s + (−0.125 − 0.216i)16-s + (−1.09 − 0.633i)17-s + (−0.0462 − 0.705i)18-s + (−0.421 − 0.729i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.298 + 0.954i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.298 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.298 + 0.954i$
Analytic conductor: \(8.01091\)
Root analytic conductor: \(2.83035\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1),\ 0.298 + 0.954i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.52659 - 1.85766i\)
\(L(\frac12)\) \(\approx\) \(2.52659 - 1.85766i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 + (-2.54 + 1.58i)T \)
7 \( 1 \)
good5 \( 1 + (-0.790 + 0.456i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-12.6 - 7.27i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 0.583T + 169T^{2} \)
17 \( 1 + (18.6 + 10.7i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (8 + 13.8i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-33.6 + 19.4i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 35.7iT - 841T^{2} \)
31 \( 1 + (29.2 - 50.6i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (10 + 17.3i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 8.75iT - 1.68e3T^{2} \)
43 \( 1 + 11.7T + 1.84e3T^{2} \)
47 \( 1 + (-7.34 + 4.24i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-44.0 - 25.4i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-50.4 - 29.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-19.4 - 33.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (35.2 - 61.1i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 17.0iT - 5.04e3T^{2} \)
73 \( 1 + (-36.1 + 62.6i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (10.4 + 18.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 145. iT - 6.88e3T^{2} \)
89 \( 1 + (46.4 - 26.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 111.T + 9.40e3T^{2} \)
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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

−11.55336105525568390272303213218, −10.55957399537955952760011930052, −9.111519195367766342466045758256, −8.932989168759032369789009995466, −7.03522069932577495039934018786, −6.79756924465124349189034960758, −5.06587292709086971939106101615, −3.92361830718743485591891101843, −2.67512947712054774424937063375, −1.40552595686447402707887855025, 2.10369322919487218932268070442, 3.59003693117319139788510754062, 4.30500516719907426906386759388, 5.74820298584236155327597510232, 6.75947581811643115345291466135, 7.989170523801049980286608650900, 8.822847381252210457567832794717, 9.700842552004208506407858974545, 10.90437678100174080370040075834, 11.69719902394466424447211068415

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