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Properties

Label 2-294-7.6-c2-0-1
Degree $2$
Conductor $294$
Sign $-0.654 - 0.755i$
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  − 1.41·2-s + 1.73i·3-s + 2.00·4-s + 1.43i·5-s − 2.44i·6-s − 2.82·8-s − 2.99·9-s − 2.02i·10-s + 6·11-s + 3.46i·12-s + 21.3i·13-s − 2.48·15-s + 4.00·16-s − 8.95i·17-s + 4.24·18-s + 7.22i·19-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577i·3-s + 0.500·4-s + 0.286i·5-s − 0.408i·6-s − 0.353·8-s − 0.333·9-s − 0.202i·10-s + 0.545·11-s + 0.288i·12-s + 1.64i·13-s − 0.165·15-s + 0.250·16-s − 0.526i·17-s + 0.235·18-s + 0.380i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.654 - 0.755i$
Analytic conductor: \(8.01091\)
Root analytic conductor: \(2.83035\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1),\ -0.654 - 0.755i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.350394 + 0.766978i\)
\(L(\frac12)\) \(\approx\) \(0.350394 + 0.766978i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 - 1.73iT \)
7 \( 1 \)
good5 \( 1 - 1.43iT - 25T^{2} \)
11 \( 1 - 6T + 121T^{2} \)
13 \( 1 - 21.3iT - 169T^{2} \)
17 \( 1 + 8.95iT - 289T^{2} \)
19 \( 1 - 7.22iT - 361T^{2} \)
23 \( 1 + 37.4T + 529T^{2} \)
29 \( 1 + 33.9T + 841T^{2} \)
31 \( 1 - 44.1iT - 961T^{2} \)
37 \( 1 + 27.9T + 1.36e3T^{2} \)
41 \( 1 - 54.8iT - 1.68e3T^{2} \)
43 \( 1 + 1.48T + 1.84e3T^{2} \)
47 \( 1 - 43.0iT - 2.20e3T^{2} \)
53 \( 1 + 85.4T + 2.80e3T^{2} \)
59 \( 1 + 41.2iT - 3.48e3T^{2} \)
61 \( 1 - 1.18iT - 3.72e3T^{2} \)
67 \( 1 - 4.39T + 4.48e3T^{2} \)
71 \( 1 - 137.T + 5.04e3T^{2} \)
73 \( 1 - 78.9iT - 5.32e3T^{2} \)
79 \( 1 - 98.3T + 6.24e3T^{2} \)
83 \( 1 + 110. iT - 6.88e3T^{2} \)
89 \( 1 - 20.7iT - 7.92e3T^{2} \)
97 \( 1 - 10.9iT - 9.40e3T^{2} \)
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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

−11.59975649560990528932832937899, −10.93548872324856347514920940744, −9.817101555391021170333751292631, −9.250915128201347631082817172552, −8.266277398457491535879011497439, −7.03143213549751254061718245453, −6.19875004119198641798718502491, −4.68397621822193774538568887843, −3.45067551127870137981916210702, −1.81273904851555946954955418094, 0.51122406546412885855857359180, 2.06351474936104447191572524978, 3.62629660272084183706746554365, 5.42628072548673656266152104409, 6.37149709895808899239751472455, 7.57950531500514982464906056569, 8.240609907399805199154632531170, 9.227154139666108507347285340545, 10.26316893891593790474512465312, 11.11989647772339812057496917707

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