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Properties

Label 2-414-23.6-c1-0-3
Degree $2$
Conductor $414$
Sign $-0.217 - 0.975i$
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 + (−1.27 + 0.818i)5-s + (0.369 − 2.56i)7-s + (0.959 − 0.281i)8-s + (−0.215 − 1.49i)10-s + (2.08 + 4.55i)11-s + (0.686 + 4.77i)13-s + (2.18 + 1.40i)14-s + (−0.142 + 0.989i)16-s + (−0.565 + 0.652i)17-s + (4.58 + 5.29i)19-s + (1.45 + 0.426i)20-s − 5.00·22-s + (−4.66 − 1.11i)23-s + ⋯
L(s)  = 1  + (−0.293 + 0.643i)2-s + (−0.327 − 0.377i)4-s + (−0.569 + 0.365i)5-s + (0.139 − 0.970i)7-s + (0.339 − 0.0996i)8-s + (−0.0681 − 0.473i)10-s + (0.627 + 1.37i)11-s + (0.190 + 1.32i)13-s + (0.583 + 0.374i)14-s + (−0.0355 + 0.247i)16-s + (−0.137 + 0.158i)17-s + (1.05 + 1.21i)19-s + (0.324 + 0.0953i)20-s − 1.06·22-s + (−0.972 − 0.231i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.217 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.217 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $-0.217 - 0.975i$
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: \((2,\ 414,\ (\ :1/2),\ -0.217 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.618047 + 0.771233i\)
\(L(\frac12)\) \(\approx\) \(0.618047 + 0.771233i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.66 + 1.11i)T \)
good5 \( 1 + (1.27 - 0.818i)T + (2.07 - 4.54i)T^{2} \)
7 \( 1 + (-0.369 + 2.56i)T + (-6.71 - 1.97i)T^{2} \)
11 \( 1 + (-2.08 - 4.55i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (-0.686 - 4.77i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (0.565 - 0.652i)T + (-2.41 - 16.8i)T^{2} \)
19 \( 1 + (-4.58 - 5.29i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (-2.03 + 2.35i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (-1.12 + 0.330i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (0.506 + 0.325i)T + (15.3 + 33.6i)T^{2} \)
41 \( 1 + (4.45 - 2.86i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (1.40 + 0.411i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 - 12.6T + 47T^{2} \)
53 \( 1 + (-0.602 + 4.18i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (-0.566 - 3.94i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (-6.96 + 2.04i)T + (51.3 - 32.9i)T^{2} \)
67 \( 1 + (6.70 - 14.6i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (2.44 - 5.35i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (9.41 + 10.8i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (1.47 + 10.2i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (2.22 + 1.42i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (5.02 + 1.47i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (-12.5 + 8.08i)T + (40.2 - 88.2i)T^{2} \)
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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

−11.57880212330844608201321024500, −10.28249823418655235809754227991, −9.763378249215005891524524478036, −8.633505505512749998171997990978, −7.42786527524457995215005796846, −7.15968879599965058086886083593, −6.02303826054685892674354687041, −4.45547295049211555024961021453, −3.87125174881451287665471990994, −1.63449829283500929252582960066, 0.77022700189949337020248918734, 2.70086642688814244968025445140, 3.68684077998307807466271089964, 5.10608643954446410182860203438, 6.03016552058143061680275625141, 7.55897984060965866501860588067, 8.540559237631395076490136030283, 8.925374374078667320564671366250, 10.12488250390218305304134784278, 11.13448783933823501943331379575

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