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Properties

Label 2-414-23.13-c1-0-4
Degree $2$
Conductor $414$
Sign $0.746 - 0.665i$
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  + (0.654 + 0.755i)2-s + (−0.142 + 0.989i)4-s + (0.355 − 0.779i)5-s + (2.70 + 0.795i)7-s + (−0.841 + 0.540i)8-s + (0.822 − 0.241i)10-s + (3.69 − 4.26i)11-s + (−0.549 + 0.161i)13-s + (1.17 + 2.56i)14-s + (−0.959 − 0.281i)16-s + (0.259 + 1.80i)17-s + (−1.09 + 7.59i)19-s + (0.720 + 0.463i)20-s + 5.63·22-s + (−4.59 + 1.36i)23-s + ⋯
L(s)  = 1  + (0.463 + 0.534i)2-s + (−0.0711 + 0.494i)4-s + (0.159 − 0.348i)5-s + (1.02 + 0.300i)7-s + (−0.297 + 0.191i)8-s + (0.259 − 0.0763i)10-s + (1.11 − 1.28i)11-s + (−0.152 + 0.0447i)13-s + (0.313 + 0.686i)14-s + (−0.239 − 0.0704i)16-s + (0.0629 + 0.437i)17-s + (−0.250 + 1.74i)19-s + (0.161 + 0.103i)20-s + 1.20·22-s + (−0.958 + 0.283i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.746 - 0.665i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ 0.746 - 0.665i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85532 + 0.706689i\)
\(L(\frac12)\) \(\approx\) \(1.85532 + 0.706689i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.654 - 0.755i)T \)
3 \( 1 \)
23 \( 1 + (4.59 - 1.36i)T \)
good5 \( 1 + (-0.355 + 0.779i)T + (-3.27 - 3.77i)T^{2} \)
7 \( 1 + (-2.70 - 0.795i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (-3.69 + 4.26i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (0.549 - 0.161i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (-0.259 - 1.80i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (1.09 - 7.59i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (0.707 + 4.92i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-6.83 + 4.39i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (-0.172 - 0.377i)T + (-24.2 + 27.9i)T^{2} \)
41 \( 1 + (-1.18 + 2.58i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (4.45 + 2.86i)T + (17.8 + 39.1i)T^{2} \)
47 \( 1 + 6.28T + 47T^{2} \)
53 \( 1 + (-0.404 - 0.118i)T + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (6.62 - 1.94i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (4.64 - 2.98i)T + (25.3 - 55.4i)T^{2} \)
67 \( 1 + (8.17 + 9.43i)T + (-9.53 + 66.3i)T^{2} \)
71 \( 1 + (2.97 + 3.42i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-0.375 + 2.60i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (4.41 - 1.29i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (6.92 + 15.1i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-10.1 - 6.51i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (1.32 - 2.90i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.72193542630031810159378696206, −10.49816953168416150326933267331, −9.279058705374167132379597780006, −8.337384730929755393675751316956, −7.84165270745517879013002135580, −6.26640807323026844328467434637, −5.75826172282562748378092561259, −4.51880061946415443589413226790, −3.52846588411862838319748469691, −1.65499168395151640187700797210, 1.51698604072744170448032570924, 2.79835370056975575458153060001, 4.42567885221491356750609996755, 4.84311902566159974537655136622, 6.47852479558423462762757969738, 7.16807265112405645945777389446, 8.480330209123605468579078822789, 9.512952801404417691120234571946, 10.34301150539347021830895010661, 11.24682365444059527404447665913

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