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Properties

Label 2-414-23.2-c1-0-3
Degree $2$
Conductor $414$
Sign $0.999 - 0.00758i$
Analytic cond. $3.30580$
Root an. cond. $1.81818$
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  + (−0.841 − 0.540i)2-s + (0.415 + 0.909i)4-s + (3.26 + 0.957i)5-s + (−0.297 + 0.342i)7-s + (0.142 − 0.989i)8-s + (−2.22 − 2.56i)10-s + (2.53 − 1.63i)11-s + (−0.592 − 0.683i)13-s + (0.435 − 0.127i)14-s + (−0.654 + 0.755i)16-s + (−2.61 + 5.72i)17-s + (2.73 + 5.99i)19-s + (0.483 + 3.36i)20-s − 3.01·22-s + (0.838 − 4.72i)23-s + ⋯
L(s)  = 1  + (−0.594 − 0.382i)2-s + (0.207 + 0.454i)4-s + (1.45 + 0.428i)5-s + (−0.112 + 0.129i)7-s + (0.0503 − 0.349i)8-s + (−0.703 − 0.812i)10-s + (0.765 − 0.491i)11-s + (−0.164 − 0.189i)13-s + (0.116 − 0.0341i)14-s + (−0.163 + 0.188i)16-s + (−0.634 + 1.38i)17-s + (0.627 + 1.37i)19-s + (0.108 + 0.752i)20-s − 0.643·22-s + (0.174 − 0.984i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.999 - 0.00758i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ 0.999 - 0.00758i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33966 + 0.00508169i\)
\(L(\frac12)\) \(\approx\) \(1.33966 + 0.00508169i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.841 + 0.540i)T \)
3 \( 1 \)
23 \( 1 + (-0.838 + 4.72i)T \)
good5 \( 1 + (-3.26 - 0.957i)T + (4.20 + 2.70i)T^{2} \)
7 \( 1 + (0.297 - 0.342i)T + (-0.996 - 6.92i)T^{2} \)
11 \( 1 + (-2.53 + 1.63i)T + (4.56 - 10.0i)T^{2} \)
13 \( 1 + (0.592 + 0.683i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (2.61 - 5.72i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (-2.73 - 5.99i)T + (-12.4 + 14.3i)T^{2} \)
29 \( 1 + (-0.645 + 1.41i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (-1.07 + 7.47i)T + (-29.7 - 8.73i)T^{2} \)
37 \( 1 + (1.19 - 0.349i)T + (31.1 - 20.0i)T^{2} \)
41 \( 1 + (-7.37 - 2.16i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (0.887 + 6.16i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 + (6.13 - 7.07i)T + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (4.43 + 5.11i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (0.447 - 3.11i)T + (-58.5 - 17.1i)T^{2} \)
67 \( 1 + (5.96 + 3.83i)T + (27.8 + 60.9i)T^{2} \)
71 \( 1 + (10.6 + 6.87i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (0.587 + 1.28i)T + (-47.8 + 55.1i)T^{2} \)
79 \( 1 + (-0.618 - 0.713i)T + (-11.2 + 78.1i)T^{2} \)
83 \( 1 + (7.57 - 2.22i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (1.58 + 11.0i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + (1.40 + 0.412i)T + (81.6 + 52.4i)T^{2} \)
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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

−10.87355484217257244595702958732, −10.32591900062249457363534577460, −9.482145230083215715824959459896, −8.760305662645022079502935682860, −7.64186952100382432475570001947, −6.23582233012290455864540058520, −5.94583404790487389820106972217, −4.08863163615783666530885086605, −2.69249484930639750590753777668, −1.55698407170119570870948936919, 1.29574278207859576194667946467, 2.65762510125728544523033178815, 4.70938317720748797684833463082, 5.52456991042273689475645491894, 6.70746138091341104924061628929, 7.26869846404143589134774898214, 8.928974335167376116465666434257, 9.282533180848268845411643363123, 9.951099995184200033056236115091, 11.07184846817038739581546785589

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