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Properties

Label 2-414-23.8-c1-0-3
Degree $2$
Conductor $414$
Sign $0.983 - 0.182i$
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 + (1.66 + 1.92i)5-s + (1.75 + 1.12i)7-s + (−0.415 + 0.909i)8-s + (2.14 − 1.37i)10-s + (0.543 + 3.77i)11-s + (−5.21 + 3.34i)13-s + (1.36 − 1.57i)14-s + (0.841 + 0.540i)16-s + (1.24 − 0.366i)17-s + (2.37 + 0.698i)19-s + (−1.05 − 2.31i)20-s + 3.81·22-s + (1.33 − 4.60i)23-s + ⋯
L(s)  = 1  + (0.100 − 0.699i)2-s + (−0.479 − 0.140i)4-s + (0.745 + 0.860i)5-s + (0.663 + 0.426i)7-s + (−0.146 + 0.321i)8-s + (0.677 − 0.435i)10-s + (0.163 + 1.13i)11-s + (−1.44 + 0.929i)13-s + (0.365 − 0.421i)14-s + (0.210 + 0.135i)16-s + (0.303 − 0.0890i)17-s + (0.546 + 0.160i)19-s + (−0.236 − 0.517i)20-s + 0.814·22-s + (0.278 − 0.960i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.983 - 0.182i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ 0.983 - 0.182i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56661 + 0.144017i\)
\(L(\frac12)\) \(\approx\) \(1.56661 + 0.144017i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.33 + 4.60i)T \)
good5 \( 1 + (-1.66 - 1.92i)T + (-0.711 + 4.94i)T^{2} \)
7 \( 1 + (-1.75 - 1.12i)T + (2.90 + 6.36i)T^{2} \)
11 \( 1 + (-0.543 - 3.77i)T + (-10.5 + 3.09i)T^{2} \)
13 \( 1 + (5.21 - 3.34i)T + (5.40 - 11.8i)T^{2} \)
17 \( 1 + (-1.24 + 0.366i)T + (14.3 - 9.19i)T^{2} \)
19 \( 1 + (-2.37 - 0.698i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (-6.87 + 2.01i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (1.67 - 3.65i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (-7.48 + 8.63i)T + (-5.26 - 36.6i)T^{2} \)
41 \( 1 + (2.81 + 3.24i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (1.15 + 2.52i)T + (-28.1 + 32.4i)T^{2} \)
47 \( 1 + 9.34T + 47T^{2} \)
53 \( 1 + (-1.99 - 1.27i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (0.514 - 0.330i)T + (24.5 - 53.6i)T^{2} \)
61 \( 1 + (-1.67 + 3.67i)T + (-39.9 - 46.1i)T^{2} \)
67 \( 1 + (1.82 - 12.6i)T + (-64.2 - 18.8i)T^{2} \)
71 \( 1 + (0.940 - 6.54i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-2.80 - 0.824i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (1.58 - 1.02i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (-6.83 + 7.89i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (2.08 + 4.55i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (8.21 + 9.48i)T + (-13.8 + 96.0i)T^{2} \)
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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

−11.32216164243841076080689783697, −10.14692275594502765257214010470, −9.829521591976069244147010812774, −8.763604533315460386215449539652, −7.43018877013621799159291127568, −6.58674991730623551457130229737, −5.23441391449983421830900783373, −4.39914053793232901882122169736, −2.68142412760329393246235911461, −1.94864894320058365140896299816, 1.10520670132275345197646747597, 3.11840213513718643718502735159, 4.79709643410867687487266511059, 5.29260825023413795739528588999, 6.34863556638180556474490959756, 7.67793789141899284320715007726, 8.228136615696002037066801579681, 9.366099907096482357270038769721, 10.00183304498455976616431974609, 11.23182407806104789256456990638

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