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Properties

Label 2-294-21.11-c2-0-23
Degree $2$
Conductor $294$
Sign $-0.0156 + 0.999i$
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 + (2.28 − 1.94i)3-s + (0.999 − 1.73i)4-s + (3.46 − 2i)5-s + (1.41 − 4i)6-s − 2.82i·8-s + (1.39 − 8.89i)9-s + (2.82 − 4.89i)10-s + (−8.57 − 4.94i)11-s + (−1.09 − 5.89i)12-s − 12.7·13-s + (3.99 − 11.3i)15-s + (−2.00 − 3.46i)16-s + (27.7 + 16i)17-s + (−4.57 − 11.8i)18-s + (14.1 + 24.4i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.760 − 0.649i)3-s + (0.249 − 0.433i)4-s + (0.692 − 0.400i)5-s + (0.235 − 0.666i)6-s − 0.353i·8-s + (0.155 − 0.987i)9-s + (0.282 − 0.489i)10-s + (−0.779 − 0.449i)11-s + (−0.0913 − 0.491i)12-s − 0.979·13-s + (0.266 − 0.754i)15-s + (−0.125 − 0.216i)16-s + (1.63 + 0.941i)17-s + (−0.254 − 0.659i)18-s + (0.744 + 1.28i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0156 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0156 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.0156 + 0.999i$
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: \((2,\ 294,\ (\ :1),\ -0.0156 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.17143 - 2.20561i\)
\(L(\frac12)\) \(\approx\) \(2.17143 - 2.20561i\)
\(L(2)\) 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 \( 1 + (-2.28 + 1.94i)T \)
7 \( 1 \)
good5 \( 1 + (-3.46 + 2i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (8.57 + 4.94i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 12.7T + 169T^{2} \)
17 \( 1 + (-27.7 - 16i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-14.1 - 24.4i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (1.22 - 0.707i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 56.5iT - 841T^{2} \)
31 \( 1 + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-25 - 43.3i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 16iT - 1.68e3T^{2} \)
43 \( 1 + 52T + 1.84e3T^{2} \)
47 \( 1 + (-29.4 + 17i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-39.1 - 22.6i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-27.7 - 16i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (16.2 + 28.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-6 + 10.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 89.0iT - 5.04e3T^{2} \)
73 \( 1 + (16.2 - 28.1i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-20 - 34.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 62iT - 6.88e3T^{2} \)
89 \( 1 + (-48.4 + 28i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 24.0T + 9.40e3T^{2} \)
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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

−11.73448681540440930374571710324, −9.979492111125398696239282254811, −9.845456234116725444517213302582, −8.244589480376609489266304074084, −7.60469584886886967857518008378, −6.10255987939997890900609216961, −5.37643955543657188582968789519, −3.74720079452651639042016188897, −2.58396582972157380265382029560, −1.32247015308804915805013519530, 2.41326479099061904716751098225, 3.24128203627731604976299490836, 4.86731093505699558186272934943, 5.41823470593050385868925764746, 7.11496930770432269458767642909, 7.68085781594480221668733084767, 9.093169461404310937988070932857, 9.888851918380060003879917856485, 10.64222836916433273251608069023, 11.93378063218238948065401827082

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