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Properties

Label 2-414-207.11-c1-0-17
Degree $2$
Conductor $414$
Sign $-0.469 + 0.882i$
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  + (−0.998 + 0.0475i)2-s + (−0.282 + 1.70i)3-s + (0.995 − 0.0950i)4-s + (0.132 + 0.125i)5-s + (0.200 − 1.72i)6-s + (−0.681 − 3.53i)7-s + (−0.989 + 0.142i)8-s + (−2.84 − 0.964i)9-s + (−0.137 − 0.119i)10-s + (−1.13 + 0.586i)11-s + (−0.118 + 1.72i)12-s + (−6.21 − 1.19i)13-s + (0.849 + 3.50i)14-s + (−0.252 + 0.190i)15-s + (0.981 − 0.189i)16-s + (0.253 + 0.554i)17-s + ⋯
L(s)  = 1  + (−0.706 + 0.0336i)2-s + (−0.162 + 0.986i)3-s + (0.497 − 0.0475i)4-s + (0.0590 + 0.0563i)5-s + (0.0819 − 0.702i)6-s + (−0.257 − 1.33i)7-s + (−0.349 + 0.0503i)8-s + (−0.946 − 0.321i)9-s + (−0.0436 − 0.0377i)10-s + (−0.342 + 0.176i)11-s + (−0.0342 + 0.498i)12-s + (−1.72 − 0.332i)13-s + (0.226 + 0.935i)14-s + (−0.0652 + 0.0491i)15-s + (0.245 − 0.0473i)16-s + (0.0614 + 0.134i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.469 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $-0.469 + 0.882i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ -0.469 + 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.124027 - 0.206421i\)
\(L(\frac12)\) \(\approx\) \(0.124027 - 0.206421i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.998 - 0.0475i)T \)
3 \( 1 + (0.282 - 1.70i)T \)
23 \( 1 + (2.61 - 4.02i)T \)
good5 \( 1 + (-0.132 - 0.125i)T + (0.237 + 4.99i)T^{2} \)
7 \( 1 + (0.681 + 3.53i)T + (-6.49 + 2.60i)T^{2} \)
11 \( 1 + (1.13 - 0.586i)T + (6.38 - 8.96i)T^{2} \)
13 \( 1 + (6.21 + 1.19i)T + (12.0 + 4.83i)T^{2} \)
17 \( 1 + (-0.253 - 0.554i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (3.27 + 1.49i)T + (12.4 + 14.3i)T^{2} \)
29 \( 1 + (-0.372 + 3.89i)T + (-28.4 - 5.48i)T^{2} \)
31 \( 1 + (-1.12 + 0.886i)T + (7.30 - 30.1i)T^{2} \)
37 \( 1 + (0.286 - 0.975i)T + (-31.1 - 20.0i)T^{2} \)
41 \( 1 + (0.230 - 0.241i)T + (-1.95 - 40.9i)T^{2} \)
43 \( 1 + (0.197 - 0.251i)T + (-10.1 - 41.7i)T^{2} \)
47 \( 1 + (-2.40 - 1.39i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.06 + 6.99i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (1.09 - 5.66i)T + (-54.7 - 21.9i)T^{2} \)
61 \( 1 + (-0.669 + 1.67i)T + (-44.1 - 42.0i)T^{2} \)
67 \( 1 + (5.29 - 10.2i)T + (-38.8 - 54.5i)T^{2} \)
71 \( 1 + (-1.34 - 2.09i)T + (-29.4 + 64.5i)T^{2} \)
73 \( 1 + (-6.39 + 14.0i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (-8.92 + 3.08i)T + (62.0 - 48.8i)T^{2} \)
83 \( 1 + (13.1 - 12.4i)T + (3.94 - 82.9i)T^{2} \)
89 \( 1 + (-0.400 + 2.78i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (11.1 + 2.70i)T + (86.2 + 44.4i)T^{2} \)
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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

−10.49814083494073783940360927402, −10.09535600212425430893141550773, −9.459422489069867731409375041568, −8.158575515479469915733964173828, −7.36342813732586324451229512380, −6.28495306095104366484266813715, −4.96100015460168149465020487832, −3.98758857003094247427442978291, −2.58326388323228247649515021108, −0.18105273722903610331845939277, 1.99509338916320205142687327998, 2.81577947988378801848003377598, 5.07346221721947134637278141080, 6.02516091895334210931602262516, 6.95923026084454090504973714903, 7.88045135138016564257083986228, 8.747791333073085529228917117425, 9.513327581482376614321715231134, 10.62195252866965225206716587031, 11.66912221987661032911559916595

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