Properties

Label 2-414-9.7-c1-0-12
Degree $2$
Conductor $414$
Sign $0.432 + 0.901i$
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 + (0.365 − 1.69i)3-s + (−0.499 − 0.866i)4-s + (0.231 + 0.400i)5-s + (1.28 + 1.16i)6-s + (0.165 − 0.286i)7-s + 0.999·8-s + (−2.73 − 1.23i)9-s − 0.462·10-s + (2.28 − 3.95i)11-s + (−1.64 + 0.529i)12-s + (−0.380 − 0.658i)13-s + (0.165 + 0.286i)14-s + (0.762 − 0.244i)15-s + (−0.5 + 0.866i)16-s + 5.46·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.211 − 0.977i)3-s + (−0.249 − 0.433i)4-s + (0.103 + 0.179i)5-s + (0.523 + 0.474i)6-s + (0.0625 − 0.108i)7-s + 0.353·8-s + (−0.910 − 0.412i)9-s − 0.146·10-s + (0.688 − 1.19i)11-s + (−0.476 + 0.152i)12-s + (−0.105 − 0.182i)13-s + (0.0442 + 0.0765i)14-s + (0.196 − 0.0632i)15-s + (−0.125 + 0.216i)16-s + 1.32·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.432 + 0.901i$
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: \((2,\ 414,\ (\ :1/2),\ 0.432 + 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.961050 - 0.605011i\)
\(L(\frac12)\) \(\approx\) \(0.961050 - 0.605011i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.365 + 1.69i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-0.231 - 0.400i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.165 + 0.286i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.28 + 3.95i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.380 + 0.658i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.46T + 17T^{2} \)
19 \( 1 + 8.02T + 19T^{2} \)
29 \( 1 + (-3.24 + 5.62i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.49 + 7.78i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.26T + 37T^{2} \)
41 \( 1 + (-3.11 - 5.39i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.391 - 0.678i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.79 + 6.57i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.47T + 53T^{2} \)
59 \( 1 + (-6.14 - 10.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.55 - 9.61i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.83 - 11.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.24T + 71T^{2} \)
73 \( 1 + 9.95T + 73T^{2} \)
79 \( 1 + (-1.76 + 3.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.95 + 6.84i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 1.01T + 89T^{2} \)
97 \( 1 + (1.08 - 1.87i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.07085218237249561772021048659, −10.05611101729119139524425770311, −8.877997507602535848553749112059, −8.233195105577668973520444773597, −7.42795376627144161828966309835, −6.22240907301420646198189635758, −5.92739530617229548939175162145, −4.10345573663123232588973616543, −2.56307929554595840624920767424, −0.849478866672950339283296662898, 1.86138002185800806279822413499, 3.32729291849535757563414323756, 4.35849565191758322227537266352, 5.27783613229160075058067771684, 6.78278690812690403602266349352, 8.027239418471244710072734230373, 8.999539146303558661299187162241, 9.552064106776006776484100895139, 10.46136873611879185071505388490, 11.08599172818384657996699169661

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