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Properties

Label 2-414-9.7-c1-0-9
Degree $2$
Conductor $414$
Sign $0.801 - 0.597i$
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  + (0.5 − 0.866i)2-s + (1.07 + 1.35i)3-s + (−0.499 − 0.866i)4-s + (1.11 + 1.92i)5-s + (1.71 − 0.251i)6-s + (−0.920 + 1.59i)7-s − 0.999·8-s + (−0.690 + 2.91i)9-s + 2.22·10-s + (0.563 − 0.976i)11-s + (0.639 − 1.60i)12-s + (0.510 + 0.884i)13-s + (0.920 + 1.59i)14-s + (−1.42 + 3.57i)15-s + (−0.5 + 0.866i)16-s − 0.618·17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.620 + 0.784i)3-s + (−0.249 − 0.433i)4-s + (0.497 + 0.861i)5-s + (0.699 − 0.102i)6-s + (−0.348 + 0.602i)7-s − 0.353·8-s + (−0.230 + 0.973i)9-s + 0.703·10-s + (0.170 − 0.294i)11-s + (0.184 − 0.464i)12-s + (0.141 + 0.245i)13-s + (0.246 + 0.426i)14-s + (−0.367 + 0.924i)15-s + (−0.125 + 0.216i)16-s − 0.149·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.801 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.801 - 0.597i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ 0.801 - 0.597i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89549 + 0.628428i\)
\(L(\frac12)\) \(\approx\) \(1.89549 + 0.628428i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-1.07 - 1.35i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-1.11 - 1.92i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.920 - 1.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.563 + 0.976i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.510 - 0.884i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.618T + 17T^{2} \)
19 \( 1 - 2.53T + 19T^{2} \)
29 \( 1 + (-0.936 + 1.62i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.78 + 4.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.21T + 37T^{2} \)
41 \( 1 + (1.63 + 2.82i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.36 + 11.0i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.77 - 3.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.09T + 53T^{2} \)
59 \( 1 + (-0.660 - 1.14i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.42 + 2.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.72 + 9.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.45T + 71T^{2} \)
73 \( 1 - 8.40T + 73T^{2} \)
79 \( 1 + (-5.68 + 9.85i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.993 + 1.72i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 + (4.13 - 7.16i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.11012353716358685259430496556, −10.47152088448899287083145110806, −9.520909719392581958085158953580, −9.039651954194206409470022313482, −7.73356964253296769042132310850, −6.34252012134471006695511164885, −5.45402951495684970631945560389, −4.14327015103507987568395036256, −3.08586056539284702694250542440, −2.24826834372547435527326661480, 1.23519086375632857922026536783, 2.97866925575516916108092759497, 4.26722689206406977102294215306, 5.48314297184456063263752007148, 6.52612499878934647665706888410, 7.34836808895635612432126776265, 8.262208501823075877487751043253, 9.132711004608756330357984643218, 9.859752540508394248062979694503, 11.34209641895912717318472297246

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