Properties

Label 2-294-49.37-c1-0-3
Degree $2$
Conductor $294$
Sign $0.829 + 0.557i$
Analytic cond. $2.34760$
Root an. cond. $1.53218$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.826 − 0.563i)2-s + (−0.733 + 0.680i)3-s + (0.365 − 0.930i)4-s + (1.32 + 0.409i)5-s + (−0.222 + 0.974i)6-s + (0.662 − 2.56i)7-s + (−0.222 − 0.974i)8-s + (0.0747 − 0.997i)9-s + (1.32 − 0.409i)10-s + (−0.0332 − 0.443i)11-s + (0.365 + 0.930i)12-s + (4.28 + 2.06i)13-s + (−0.895 − 2.48i)14-s + (−1.25 + 0.602i)15-s + (−0.733 − 0.680i)16-s + (3.16 + 0.476i)17-s + ⋯
L(s)  = 1  + (0.584 − 0.398i)2-s + (−0.423 + 0.392i)3-s + (0.182 − 0.465i)4-s + (0.593 + 0.182i)5-s + (−0.0908 + 0.398i)6-s + (0.250 − 0.968i)7-s + (−0.0786 − 0.344i)8-s + (0.0249 − 0.332i)9-s + (0.419 − 0.129i)10-s + (−0.0100 − 0.133i)11-s + (0.105 + 0.268i)12-s + (1.18 + 0.572i)13-s + (−0.239 − 0.665i)14-s + (−0.322 + 0.155i)15-s + (−0.183 − 0.170i)16-s + (0.766 + 0.115i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.557i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.829 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.829 + 0.557i$
Analytic conductor: \(2.34760\)
Root analytic conductor: \(1.53218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ 0.829 + 0.557i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68171 - 0.512665i\)
\(L(\frac12)\) \(\approx\) \(1.68171 - 0.512665i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.826 + 0.563i)T \)
3 \( 1 + (0.733 - 0.680i)T \)
7 \( 1 + (-0.662 + 2.56i)T \)
good5 \( 1 + (-1.32 - 0.409i)T + (4.13 + 2.81i)T^{2} \)
11 \( 1 + (0.0332 + 0.443i)T + (-10.8 + 1.63i)T^{2} \)
13 \( 1 + (-4.28 - 2.06i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (-3.16 - 0.476i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (0.185 - 0.321i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.06 + 0.160i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (5.07 - 6.35i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (3.49 + 6.05i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.63 + 6.71i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (-1.57 - 6.89i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (1.62 - 7.12i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (7.91 - 5.39i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (3.32 - 8.46i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (-5.05 + 1.56i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (-4.36 - 11.1i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (5.63 + 9.75i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.77 + 2.22i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-1.42 - 0.974i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (-3.04 + 5.26i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.21 - 1.06i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (0.300 - 4.01i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 - 4.66T + 97T^{2} \)
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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

−11.40094281595283790983666642877, −10.94454015182391882991390248488, −10.06354173190009108698001853833, −9.176204268109038592297400725483, −7.68969648740531139303663171445, −6.44252050479814638781216879202, −5.63835054426504833661023619090, −4.37449768266870657765541067960, −3.43275117890650489891046692999, −1.49503857164037696774703178251, 1.87010232974881726511358218096, 3.49247793250402337078209655880, 5.25259765570898974681113890043, 5.68313606690910667758830220648, 6.71926916646405459566519241717, 7.967676804842241537325539460746, 8.815289952336864733117868008205, 10.02671589850094677347871748927, 11.24119535772072766263320298463, 11.96847862721507468437179700798

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