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Properties

Label 2-414-23.3-c1-0-4
Degree $2$
Conductor $414$
Sign $0.387 - 0.921i$
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  + (0.142 + 0.989i)2-s + (−0.959 + 0.281i)4-s + (−0.698 + 0.806i)5-s + (3.72 − 2.39i)7-s + (−0.415 − 0.909i)8-s + (−0.897 − 0.577i)10-s + (−0.234 + 1.63i)11-s + (4.60 + 2.95i)13-s + (2.89 + 3.34i)14-s + (0.841 − 0.540i)16-s + (3.76 + 1.10i)17-s + (−6.33 + 1.85i)19-s + (0.443 − 0.970i)20-s − 1.64·22-s + (4.78 − 0.323i)23-s + ⋯
L(s)  = 1  + (0.100 + 0.699i)2-s + (−0.479 + 0.140i)4-s + (−0.312 + 0.360i)5-s + (1.40 − 0.904i)7-s + (−0.146 − 0.321i)8-s + (−0.283 − 0.182i)10-s + (−0.0707 + 0.492i)11-s + (1.27 + 0.820i)13-s + (0.774 + 0.893i)14-s + (0.210 − 0.135i)16-s + (0.913 + 0.268i)17-s + (−1.45 + 0.426i)19-s + (0.0991 − 0.217i)20-s − 0.351·22-s + (0.997 − 0.0674i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.387 - 0.921i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ 0.387 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28035 + 0.850878i\)
\(L(\frac12)\) \(\approx\) \(1.28035 + 0.850878i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.142 - 0.989i)T \)
3 \( 1 \)
23 \( 1 + (-4.78 + 0.323i)T \)
good5 \( 1 + (0.698 - 0.806i)T + (-0.711 - 4.94i)T^{2} \)
7 \( 1 + (-3.72 + 2.39i)T + (2.90 - 6.36i)T^{2} \)
11 \( 1 + (0.234 - 1.63i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (-4.60 - 2.95i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (-3.76 - 1.10i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (6.33 - 1.85i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (6.14 + 1.80i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (1.25 + 2.74i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + (-1.44 - 1.67i)T + (-5.26 + 36.6i)T^{2} \)
41 \( 1 + (-2.67 + 3.09i)T + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (1.21 - 2.64i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 - 9.62T + 47T^{2} \)
53 \( 1 + (-1.87 + 1.20i)T + (22.0 - 48.2i)T^{2} \)
59 \( 1 + (8.20 + 5.27i)T + (24.5 + 53.6i)T^{2} \)
61 \( 1 + (0.647 + 1.41i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (1.71 + 11.9i)T + (-64.2 + 18.8i)T^{2} \)
71 \( 1 + (0.749 + 5.21i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (6.89 - 2.02i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (-2.26 - 1.45i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (-3.62 - 4.18i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (4.70 - 10.3i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (7.84 - 9.05i)T + (-13.8 - 96.0i)T^{2} \)
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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

−11.01321750945856084756117604270, −10.87021183437523002651979087463, −9.405963584160318784252483769557, −8.356389829395244111841235559802, −7.66041818673552466806874813896, −6.86003459353533087531988858492, −5.70923849966848764653724912791, −4.47011044875298611994174498778, −3.77416494499585712612279610399, −1.57617710299954473505568307594, 1.22596095804089145143305794996, 2.70068233994524138417519893791, 4.05833156384700370280458848116, 5.16876336301912674867943827545, 5.92038403370992401825872413669, 7.67681193480634911765447854659, 8.618578066228150886430953921097, 8.906529246849643789881662281008, 10.56962016260058597753132035707, 11.03369189027414947618692706924

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