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Properties

Label 2-414-9.4-c1-0-5
Degree $2$
Conductor $414$
Sign $-0.889 - 0.457i$
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 + (−0.958 + 1.44i)3-s + (−0.499 + 0.866i)4-s + (0.217 − 0.377i)5-s + (−1.72 − 0.109i)6-s + (2.31 + 4.00i)7-s − 0.999·8-s + (−1.16 − 2.76i)9-s + 0.435·10-s + (2.10 + 3.63i)11-s + (−0.769 − 1.55i)12-s + (1.60 − 2.77i)13-s + (−2.31 + 4.00i)14-s + (0.335 + 0.675i)15-s + (−0.5 − 0.866i)16-s − 5.97·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.553 + 0.832i)3-s + (−0.249 + 0.433i)4-s + (0.0973 − 0.168i)5-s + (−0.705 − 0.0445i)6-s + (0.874 + 1.51i)7-s − 0.353·8-s + (−0.387 − 0.922i)9-s + 0.137·10-s + (0.633 + 1.09i)11-s + (−0.222 − 0.447i)12-s + (0.444 − 0.769i)13-s + (−0.618 + 1.07i)14-s + (0.0865 + 0.174i)15-s + (−0.125 − 0.216i)16-s − 1.44·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.457i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $-0.889 - 0.457i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ -0.889 - 0.457i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.315792 + 1.30420i\)
\(L(\frac12)\) \(\approx\) \(0.315792 + 1.30420i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.958 - 1.44i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.217 + 0.377i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.31 - 4.00i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.10 - 3.63i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.60 + 2.77i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.97T + 17T^{2} \)
19 \( 1 + 6.45T + 19T^{2} \)
29 \( 1 + (1.77 + 3.07i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.03 + 3.52i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.64T + 37T^{2} \)
41 \( 1 + (-1.97 + 3.42i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.28 - 10.8i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.71 + 8.17i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.713T + 53T^{2} \)
59 \( 1 + (1.06 - 1.83i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.20 - 7.27i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.31 + 4.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.53T + 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 + (3.42 + 5.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.57 - 6.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + (-3.77 - 6.53i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.54676648965669455971694928812, −10.93076887850611003019241937984, −9.533212098370229928850094225591, −8.918166671515426825863752654202, −8.075141016992814292974781145104, −6.54948165231833538677126059752, −5.81637392031984359494154112204, −4.85683776006659658868272686921, −4.14920497446164269268180306456, −2.32930500956132890068337299037, 0.868000747050115453922118955093, 2.13764624967416130220729954907, 3.97536383064532456504524971833, 4.73223020959947472010318820844, 6.32331152066584169807330800445, 6.74179851518976984787004102704, 8.088681344700179684064907620677, 8.907269385806143400308139492546, 10.55238307534410192810817552242, 11.01400580584269595795456530325

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