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Properties

Label 2-414-23.3-c1-0-5
Degree $2$
Conductor $414$
Sign $0.538 + 0.842i$
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 + (−1.59 + 1.83i)5-s + (1.56 − 1.00i)7-s + (0.415 + 0.909i)8-s + (2.04 + 1.31i)10-s + (0.790 − 5.49i)11-s + (0.966 + 0.620i)13-s + (−1.21 − 1.40i)14-s + (0.841 − 0.540i)16-s + (7.20 + 2.11i)17-s + (4.29 − 1.26i)19-s + (1.01 − 2.21i)20-s − 5.55·22-s + (2.45 − 4.12i)23-s + ⋯
L(s)  = 1  + (−0.100 − 0.699i)2-s + (−0.479 + 0.140i)4-s + (−0.712 + 0.822i)5-s + (0.590 − 0.379i)7-s + (0.146 + 0.321i)8-s + (0.647 + 0.415i)10-s + (0.238 − 1.65i)11-s + (0.267 + 0.172i)13-s + (−0.324 − 0.374i)14-s + (0.210 − 0.135i)16-s + (1.74 + 0.513i)17-s + (0.985 − 0.289i)19-s + (0.225 − 0.494i)20-s − 1.18·22-s + (0.511 − 0.859i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.538 + 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.538 + 0.842i$
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: \((2,\ 414,\ (\ :1/2),\ 0.538 + 0.842i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08532 - 0.594033i\)
\(L(\frac12)\) \(\approx\) \(1.08532 - 0.594033i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.45 + 4.12i)T \)
good5 \( 1 + (1.59 - 1.83i)T + (-0.711 - 4.94i)T^{2} \)
7 \( 1 + (-1.56 + 1.00i)T + (2.90 - 6.36i)T^{2} \)
11 \( 1 + (-0.790 + 5.49i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (-0.966 - 0.620i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (-7.20 - 2.11i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (-4.29 + 1.26i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (-2.81 - 0.827i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (0.863 + 1.88i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + (5.06 + 5.84i)T + (-5.26 + 36.6i)T^{2} \)
41 \( 1 + (-0.450 + 0.520i)T + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (3.55 - 7.79i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 + 7.09T + 47T^{2} \)
53 \( 1 + (4.53 - 2.91i)T + (22.0 - 48.2i)T^{2} \)
59 \( 1 + (-3.39 - 2.18i)T + (24.5 + 53.6i)T^{2} \)
61 \( 1 + (0.157 + 0.344i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (0.420 + 2.92i)T + (-64.2 + 18.8i)T^{2} \)
71 \( 1 + (2.14 + 14.9i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (-11.0 + 3.23i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (-8.86 - 5.69i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (-6.84 - 7.89i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (4.47 - 9.80i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (-6.51 + 7.51i)T + (-13.8 - 96.0i)T^{2} \)
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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

−11.05902882992744146586542524396, −10.54978452353693291331265482218, −9.350767113918954832506581514351, −8.236400929818141218517789548234, −7.65263567340581780466417185274, −6.35754346085087564170263886815, −5.12613152203207338999359617533, −3.66083263289539157750268230233, −3.11455618174979534340085043555, −1.07068314253120569784829451242, 1.37232529429623233165718082571, 3.55513876181670124567602342033, 4.90054035245265210714874393048, 5.30774797157923542695490762412, 6.92165562092763456899398619752, 7.74192064189938021876283320289, 8.381624134354979700108427760552, 9.479874625673981268828595008907, 10.13376158755931479724559763428, 11.80137801441784121269899662960

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