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Properties

Degree $2$
Conductor $294$
Sign $0.997 - 0.0633i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 1.22i)2-s + (1.5 − 0.866i)3-s + (−0.999 − 1.73i)4-s + (−0.878 − 0.507i)5-s + 2.44i·6-s + 2.82·8-s + (1.5 − 2.59i)9-s + (1.24 − 0.717i)10-s + (5.12 + 8.87i)11-s + (−2.99 − 1.73i)12-s − 8.95i·13-s − 1.75·15-s + (−2.00 + 3.46i)16-s + (26.3 − 15.2i)17-s + (2.12 + 3.67i)18-s + (13.9 + 8.06i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.5 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.175 − 0.101i)5-s + 0.408i·6-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.124 − 0.0717i)10-s + (0.465 + 0.806i)11-s + (−0.249 − 0.144i)12-s − 0.689i·13-s − 0.117·15-s + (−0.125 + 0.216i)16-s + (1.54 − 0.894i)17-s + (0.117 + 0.204i)18-s + (0.735 + 0.424i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.997 - 0.0633i$
Motivic weight: \(2\)
Character: $\chi_{294} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1),\ 0.997 - 0.0633i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.60896 + 0.0510011i\)
\(L(\frac12)\) \(\approx\) \(1.60896 + 0.0510011i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 1.22i)T \)
3 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (0.878 + 0.507i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-5.12 - 8.87i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 8.95iT - 169T^{2} \)
17 \( 1 + (-26.3 + 15.2i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-13.9 - 8.06i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-3.36 + 5.82i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 30T + 841T^{2} \)
31 \( 1 + (-43.4 + 25.0i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (15.4 - 26.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 7.10iT - 1.68e3T^{2} \)
43 \( 1 + 74.4T + 1.84e3T^{2} \)
47 \( 1 + (-50.4 - 29.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-35.4 - 61.4i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (0.426 - 0.246i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (2.48 + 1.43i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (13.5 + 23.4i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 50.6T + 5.04e3T^{2} \)
73 \( 1 + (61.1 - 35.3i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (66.9 - 115. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 104. iT - 6.88e3T^{2} \)
89 \( 1 + (125. + 72.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 100. iT - 9.40e3T^{2} \)
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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

−11.83459920364975129257971503555, −10.11922641284902654392847094710, −9.757548484125119138277587043000, −8.456441721794619367145261575727, −7.76782770921302616225797303082, −6.90260381356856843820718693944, −5.70143346467320353274644825730, −4.46623690828482854975893324571, −2.93733282214066670633627877269, −1.06908117758609312279566108070, 1.33236585970647825780478904007, 3.05941227294398664619558387192, 3.89061378311851311645173330816, 5.34833976106795719295254590057, 6.82542589359470167849095017927, 8.022004803403390064516972442992, 8.758477605631697834034626620574, 9.735499232597920653733687104553, 10.48297813315233049027606874076, 11.59050221089565412216591654562

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