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Properties

Label 2-414-23.12-c1-0-9
Degree $2$
Conductor $414$
Sign $0.358 + 0.933i$
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 + (4.24 − 1.24i)5-s + (−2.17 − 2.50i)7-s + (−0.142 − 0.989i)8-s + (2.89 − 3.34i)10-s + (−1.70 − 1.09i)11-s + (−1.98 + 2.28i)13-s + (−3.18 − 0.934i)14-s + (−0.654 − 0.755i)16-s + (1.24 + 2.73i)17-s + (−2.97 + 6.50i)19-s + (0.629 − 4.38i)20-s − 2.02·22-s + (4.08 − 2.50i)23-s + ⋯
L(s)  = 1  + (0.594 − 0.382i)2-s + (0.207 − 0.454i)4-s + (1.89 − 0.557i)5-s + (−0.820 − 0.947i)7-s + (−0.0503 − 0.349i)8-s + (0.916 − 1.05i)10-s + (−0.512 − 0.329i)11-s + (−0.549 + 0.634i)13-s + (−0.850 − 0.249i)14-s + (−0.163 − 0.188i)16-s + (0.303 + 0.663i)17-s + (−0.681 + 1.49i)19-s + (0.140 − 0.979i)20-s − 0.430·22-s + (0.852 − 0.522i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.358 + 0.933i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ 0.358 + 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85558 - 1.27524i\)
\(L(\frac12)\) \(\approx\) \(1.85558 - 1.27524i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.08 + 2.50i)T \)
good5 \( 1 + (-4.24 + 1.24i)T + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (2.17 + 2.50i)T + (-0.996 + 6.92i)T^{2} \)
11 \( 1 + (1.70 + 1.09i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (1.98 - 2.28i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (-1.24 - 2.73i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (2.97 - 6.50i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (0.633 + 1.38i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (-0.822 - 5.72i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (-1.77 - 0.520i)T + (31.1 + 20.0i)T^{2} \)
41 \( 1 + (1.02 - 0.299i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (-0.268 + 1.87i)T + (-41.2 - 12.1i)T^{2} \)
47 \( 1 - 0.748T + 47T^{2} \)
53 \( 1 + (-8.36 - 9.65i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (-1.17 + 1.35i)T + (-8.39 - 58.3i)T^{2} \)
61 \( 1 + (0.556 + 3.86i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (3.11 - 2.00i)T + (27.8 - 60.9i)T^{2} \)
71 \( 1 + (-3.36 + 2.16i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (2.24 - 4.91i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (5.14 - 5.94i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (3.87 + 1.13i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (0.796 - 5.54i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (12.6 - 3.72i)T + (81.6 - 52.4i)T^{2} \)
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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

−10.69425956873498943974091713910, −10.23091446523354494377398575247, −9.582877429475742131225265997941, −8.525527343925569535241948282768, −6.90993444942350385484180521939, −6.12499556403505461489446654306, −5.30993180009168697468430491195, −4.12704491845741825417373681283, −2.68688006989285095687664688420, −1.41879956509984609710647980626, 2.42325524338771336806211417188, 2.89059652124411993885115795738, 5.04077288305126811380800901257, 5.62903764752129071924665039244, 6.51124344113919280438741101609, 7.28761238425435156273711676042, 8.916038137216905441114719048506, 9.562921533341730933249719139630, 10.34142150556186690721970704422, 11.41830936474894942196815905773

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