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Properties

Label 2-414-23.4-c1-0-5
Degree $2$
Conductor $414$
Sign $0.402 + 0.915i$
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 + (1.89 + 1.21i)5-s + (−0.521 − 3.62i)7-s + (0.959 + 0.281i)8-s + (0.319 − 2.22i)10-s + (−0.0714 + 0.156i)11-s + (0.115 − 0.802i)13-s + (−3.08 + 1.97i)14-s + (−0.142 − 0.989i)16-s + (3.89 + 4.49i)17-s + (4.75 − 5.48i)19-s + (−2.15 + 0.633i)20-s + 0.172·22-s + (3.10 − 3.65i)23-s + ⋯
L(s)  = 1  + (−0.293 − 0.643i)2-s + (−0.327 + 0.377i)4-s + (0.845 + 0.543i)5-s + (−0.196 − 1.36i)7-s + (0.339 + 0.0996i)8-s + (0.101 − 0.703i)10-s + (−0.0215 + 0.0471i)11-s + (0.0319 − 0.222i)13-s + (−0.823 + 0.529i)14-s + (−0.0355 − 0.247i)16-s + (0.944 + 1.09i)17-s + (1.08 − 1.25i)19-s + (−0.482 + 0.141i)20-s + 0.0366·22-s + (0.646 − 0.762i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.402 + 0.915i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ 0.402 + 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10822 - 0.723738i\)
\(L(\frac12)\) \(\approx\) \(1.10822 - 0.723738i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.10 + 3.65i)T \)
good5 \( 1 + (-1.89 - 1.21i)T + (2.07 + 4.54i)T^{2} \)
7 \( 1 + (0.521 + 3.62i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (0.0714 - 0.156i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (-0.115 + 0.802i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (-3.89 - 4.49i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (-4.75 + 5.48i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (0.544 + 0.627i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-3.65 - 1.07i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (-1.24 + 0.801i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (8.02 + 5.15i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (10.6 - 3.13i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 - 2.95T + 47T^{2} \)
53 \( 1 + (-1.58 - 11.0i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-0.772 + 5.36i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (-1.16 - 0.341i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (-3.43 - 7.53i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (-4.73 - 10.3i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (0.899 - 1.03i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (-0.923 + 6.42i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (5.16 - 3.31i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-0.247 + 0.0726i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (-11.7 - 7.58i)T + (40.2 + 88.2i)T^{2} \)
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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

−10.72799360092589219052142810676, −10.27119808418730099232277652851, −9.609562181462387745712118461478, −8.394930035903735856475733769799, −7.31156209647902310653523862292, −6.50112943465134077926029995976, −5.13469332249409462851100352504, −3.81676049055344411443035310911, −2.74720388555762525177975211253, −1.11786763832898018089009956113, 1.58598560259630383153744486651, 3.19369621831014243988893709531, 5.25177324707132998129068769129, 5.44844913638705965239263992131, 6.58519702735557165649412887076, 7.80274019992121272801941922682, 8.715655140831133358511414049932, 9.599159129837562491831051608586, 9.904117746023217609366227259740, 11.59932790736393558843972156915

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