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Properties

Label 2-414-23.6-c1-0-2
Degree $2$
Conductor $414$
Sign $0.609 - 0.792i$
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  + (0.415 − 0.909i)2-s + (−0.654 − 0.755i)4-s + (−2.47 + 1.59i)5-s + (−0.151 + 1.05i)7-s + (−0.959 + 0.281i)8-s + (0.418 + 2.91i)10-s + (1.74 + 3.81i)11-s + (0.834 + 5.80i)13-s + (0.897 + 0.577i)14-s + (−0.142 + 0.989i)16-s + (3.51 − 4.05i)17-s + (−1.59 − 1.83i)19-s + (2.82 + 0.829i)20-s + 4.19·22-s + (2.13 + 4.29i)23-s + ⋯
L(s)  = 1  + (0.293 − 0.643i)2-s + (−0.327 − 0.377i)4-s + (−1.10 + 0.711i)5-s + (−0.0574 + 0.399i)7-s + (−0.339 + 0.0996i)8-s + (0.132 + 0.921i)10-s + (0.525 + 1.15i)11-s + (0.231 + 1.61i)13-s + (0.239 + 0.154i)14-s + (−0.0355 + 0.247i)16-s + (0.852 − 0.983i)17-s + (−0.365 − 0.421i)19-s + (0.631 + 0.185i)20-s + 0.894·22-s + (0.445 + 0.895i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.609 - 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.609 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.609 - 0.792i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ 0.609 - 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.970450 + 0.477642i\)
\(L(\frac12)\) \(\approx\) \(0.970450 + 0.477642i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.415 + 0.909i)T \)
3 \( 1 \)
23 \( 1 + (-2.13 - 4.29i)T \)
good5 \( 1 + (2.47 - 1.59i)T + (2.07 - 4.54i)T^{2} \)
7 \( 1 + (0.151 - 1.05i)T + (-6.71 - 1.97i)T^{2} \)
11 \( 1 + (-1.74 - 3.81i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (-0.834 - 5.80i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (-3.51 + 4.05i)T + (-2.41 - 16.8i)T^{2} \)
19 \( 1 + (1.59 + 1.83i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (5.90 - 6.81i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (4.40 - 1.29i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (7.22 + 4.64i)T + (15.3 + 33.6i)T^{2} \)
41 \( 1 + (-3.67 + 2.36i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (-0.521 - 0.153i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 - 1.61T + 47T^{2} \)
53 \( 1 + (-0.409 + 2.85i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (-0.0614 - 0.427i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (-1.04 + 0.305i)T + (51.3 - 32.9i)T^{2} \)
67 \( 1 + (1.19 - 2.61i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (-2.66 + 5.84i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (-8.77 - 10.1i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (-0.186 - 1.29i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (10.9 + 7.04i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (-6.91 - 2.02i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (-9.65 + 6.20i)T + (40.2 - 88.2i)T^{2} \)
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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

−11.48456881017841826445674179934, −10.75681245601528433540706973072, −9.468254590721193620355948428491, −8.967592814994292194334412607828, −7.30676668765469912659018729996, −6.97971296110070128289128545070, −5.35778408439113461910696279048, −4.19495842919904058708291621265, −3.37465477958789807756286491374, −1.90621843097277047063820267114, 0.66403404568143706696033044759, 3.42373810581689663602888372028, 4.05856895741895308224798494162, 5.40430598047216029951800586215, 6.21299673978041225414563419004, 7.63610531494960427541925700219, 8.152339122457462748684921592358, 8.852306497242009608809178589788, 10.27664170861971484939132564842, 11.14566697325364170296781017112

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