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Properties

Label 2-294-7.6-c4-0-7
Degree $2$
Conductor $294$
Sign $-0.987 + 0.156i$
Analytic cond. $30.3907$
Root an. cond. $5.51278$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.82·2-s + 5.19i·3-s + 8.00·4-s + 43.8i·5-s − 14.6i·6-s − 22.6·8-s − 27·9-s − 123. i·10-s + 66.2·11-s + 41.5i·12-s + 273. i·13-s − 227.·15-s + 64.0·16-s + 118. i·17-s + 76.3·18-s + 588. i·19-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577i·3-s + 0.500·4-s + 1.75i·5-s − 0.408i·6-s − 0.353·8-s − 0.333·9-s − 1.23i·10-s + 0.547·11-s + 0.288i·12-s + 1.61i·13-s − 1.01·15-s + 0.250·16-s + 0.409i·17-s + 0.235·18-s + 1.63i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.156i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.987 + 0.156i$
Analytic conductor: \(30.3907\)
Root analytic conductor: \(5.51278\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :2),\ -0.987 + 0.156i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.182750410\)
\(L(\frac12)\) \(\approx\) \(1.182750410\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.82T \)
3 \( 1 - 5.19iT \)
7 \( 1 \)
good5 \( 1 - 43.8iT - 625T^{2} \)
11 \( 1 - 66.2T + 1.46e4T^{2} \)
13 \( 1 - 273. iT - 2.85e4T^{2} \)
17 \( 1 - 118. iT - 8.35e4T^{2} \)
19 \( 1 - 588. iT - 1.30e5T^{2} \)
23 \( 1 - 157.T + 2.79e5T^{2} \)
29 \( 1 - 1.60e3T + 7.07e5T^{2} \)
31 \( 1 + 592. iT - 9.23e5T^{2} \)
37 \( 1 - 100.T + 1.87e6T^{2} \)
41 \( 1 - 1.78e3iT - 2.82e6T^{2} \)
43 \( 1 - 22.2T + 3.41e6T^{2} \)
47 \( 1 + 2.45e3iT - 4.87e6T^{2} \)
53 \( 1 + 165.T + 7.89e6T^{2} \)
59 \( 1 + 5.84e3iT - 1.21e7T^{2} \)
61 \( 1 - 7.07e3iT - 1.38e7T^{2} \)
67 \( 1 + 3.60e3T + 2.01e7T^{2} \)
71 \( 1 + 4.38e3T + 2.54e7T^{2} \)
73 \( 1 + 8.65e3iT - 2.83e7T^{2} \)
79 \( 1 - 227.T + 3.89e7T^{2} \)
83 \( 1 + 1.15e4iT - 4.74e7T^{2} \)
89 \( 1 + 370. iT - 6.27e7T^{2} \)
97 \( 1 - 1.20e4iT - 8.85e7T^{2} \)
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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

−11.46963598498731009955076062972, −10.43528416688274974332908524385, −9.991003183627987068156663374730, −8.930717017697084424306321599233, −7.78409769983555358489945181358, −6.68139716155003345111255791678, −6.15330913310433080614590975124, −4.19664285228241632944072040260, −3.13838497265856133027044596014, −1.83987933684413202820093964641, 0.53387331261307044737256866574, 1.14629093394430805580474628204, 2.80941997527871219795346169104, 4.67160441109647762343515798254, 5.60684500571733468419133631757, 6.90071991057177308852788326523, 8.014977881224965730785246838796, 8.695148875849173026755288758226, 9.383008539379129028980691429070, 10.57047386399546491201234278632

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