Properties

Label 2-414-207.13-c1-0-14
Degree $2$
Conductor $414$
Sign $0.156 + 0.987i$
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  + (0.327 − 0.945i)2-s + (−1.08 + 1.35i)3-s + (−0.786 − 0.618i)4-s + (0.204 + 0.287i)5-s + (0.923 + 1.46i)6-s + (−0.726 − 2.99i)7-s + (−0.841 + 0.540i)8-s + (−0.657 − 2.92i)9-s + (0.338 − 0.0992i)10-s + (4.81 + 0.927i)11-s + (1.68 − 0.394i)12-s + (−0.235 + 0.971i)13-s + (−3.06 − 0.292i)14-s + (−0.609 − 0.0342i)15-s + (0.235 + 0.971i)16-s + (−0.863 − 6.00i)17-s + ⋯
L(s)  = 1  + (0.231 − 0.668i)2-s + (−0.624 + 0.780i)3-s + (−0.393 − 0.309i)4-s + (0.0913 + 0.128i)5-s + (0.377 + 0.598i)6-s + (−0.274 − 1.13i)7-s + (−0.297 + 0.191i)8-s + (−0.219 − 0.975i)9-s + (0.106 − 0.0313i)10-s + (1.45 + 0.279i)11-s + (0.486 − 0.113i)12-s + (−0.0653 + 0.269i)13-s + (−0.819 − 0.0782i)14-s + (−0.157 − 0.00883i)15-s + (0.0589 + 0.242i)16-s + (−0.209 − 1.45i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.156 + 0.987i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ 0.156 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.879009 - 0.750788i\)
\(L(\frac12)\) \(\approx\) \(0.879009 - 0.750788i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.327 + 0.945i)T \)
3 \( 1 + (1.08 - 1.35i)T \)
23 \( 1 + (1.29 + 4.61i)T \)
good5 \( 1 + (-0.204 - 0.287i)T + (-1.63 + 4.72i)T^{2} \)
7 \( 1 + (0.726 + 2.99i)T + (-6.22 + 3.20i)T^{2} \)
11 \( 1 + (-4.81 - 0.927i)T + (10.2 + 4.08i)T^{2} \)
13 \( 1 + (0.235 - 0.971i)T + (-11.5 - 5.95i)T^{2} \)
17 \( 1 + (0.863 + 6.00i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (-0.209 + 1.45i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (-7.50 + 5.90i)T + (6.83 - 28.1i)T^{2} \)
31 \( 1 + (-0.0965 - 2.02i)T + (-30.8 + 2.94i)T^{2} \)
37 \( 1 + (-3.58 - 7.85i)T + (-24.2 + 27.9i)T^{2} \)
41 \( 1 + (5.48 + 7.70i)T + (-13.4 + 38.7i)T^{2} \)
43 \( 1 + (0.473 - 9.94i)T + (-42.8 - 4.08i)T^{2} \)
47 \( 1 + (0.280 + 0.486i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.36 + 2.74i)T + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (2.03 - 8.36i)T + (-52.4 - 27.0i)T^{2} \)
61 \( 1 + (3.86 + 1.99i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (9.81 - 1.89i)T + (62.2 - 24.9i)T^{2} \)
71 \( 1 + (-7.90 - 9.12i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (0.808 - 5.62i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (-6.89 - 6.57i)T + (3.75 + 78.9i)T^{2} \)
83 \( 1 + (3.18 - 4.47i)T + (-27.1 - 78.4i)T^{2} \)
89 \( 1 + (-3.18 - 2.04i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (2.38 - 0.228i)T + (95.2 - 18.3i)T^{2} \)
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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

−11.08099929910700093409159309855, −10.05104230773467382706422080372, −9.703433467197713322559816956352, −8.612374230415412037679266978537, −6.88676034549349149245607822988, −6.33833470109571983332942606132, −4.65457878556949076576562840277, −4.28374451119135734202787473042, −2.97393137722179747943357798989, −0.825064356343782921924638339085, 1.64426459912890018190076026622, 3.45892633305354960967391026782, 4.98083605044150246137471842179, 6.03903300755158412960747611909, 6.37599809356153488878200437773, 7.59583283207222176970900534020, 8.596371705072928998288738935409, 9.288685303968162014720610127203, 10.67614643075140979896172841167, 11.75581741554730256029440329064

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