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Properties

Label 2-414-23.2-c1-0-0
Degree $2$
Conductor $414$
Sign $-0.826 - 0.563i$
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  + (−0.841 − 0.540i)2-s + (0.415 + 0.909i)4-s + (−0.186 − 0.0547i)5-s + (−1.27 + 1.47i)7-s + (0.142 − 0.989i)8-s + (0.127 + 0.146i)10-s + (−5.30 + 3.40i)11-s + (−3.15 − 3.63i)13-s + (1.87 − 0.549i)14-s + (−0.654 + 0.755i)16-s + (1.33 − 2.91i)17-s + (0.357 + 0.781i)19-s + (−0.0276 − 0.192i)20-s + 6.30·22-s + (−3.37 + 3.40i)23-s + ⋯
L(s)  = 1  + (−0.594 − 0.382i)2-s + (0.207 + 0.454i)4-s + (−0.0833 − 0.0244i)5-s + (−0.483 + 0.557i)7-s + (0.0503 − 0.349i)8-s + (0.0402 + 0.0464i)10-s + (−1.59 + 1.02i)11-s + (−0.874 − 1.00i)13-s + (0.500 − 0.146i)14-s + (−0.163 + 0.188i)16-s + (0.323 − 0.707i)17-s + (0.0819 + 0.179i)19-s + (−0.00618 − 0.0429i)20-s + 1.34·22-s + (−0.704 + 0.709i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 - 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $-0.826 - 0.563i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ -0.826 - 0.563i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0500001 + 0.162157i\)
\(L(\frac12)\) \(\approx\) \(0.0500001 + 0.162157i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.841 + 0.540i)T \)
3 \( 1 \)
23 \( 1 + (3.37 - 3.40i)T \)
good5 \( 1 + (0.186 + 0.0547i)T + (4.20 + 2.70i)T^{2} \)
7 \( 1 + (1.27 - 1.47i)T + (-0.996 - 6.92i)T^{2} \)
11 \( 1 + (5.30 - 3.40i)T + (4.56 - 10.0i)T^{2} \)
13 \( 1 + (3.15 + 3.63i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (-1.33 + 2.91i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (-0.357 - 0.781i)T + (-12.4 + 14.3i)T^{2} \)
29 \( 1 + (2.25 - 4.92i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (-0.144 + 1.00i)T + (-29.7 - 8.73i)T^{2} \)
37 \( 1 + (6.61 - 1.94i)T + (31.1 - 20.0i)T^{2} \)
41 \( 1 + (0.932 + 0.273i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (-0.731 - 5.09i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 - 7.41T + 47T^{2} \)
53 \( 1 + (-1.31 + 1.51i)T + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (5.63 + 6.50i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (1.72 - 11.9i)T + (-58.5 - 17.1i)T^{2} \)
67 \( 1 + (-5.96 - 3.83i)T + (27.8 + 60.9i)T^{2} \)
71 \( 1 + (0.933 + 0.599i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (5.47 + 11.9i)T + (-47.8 + 55.1i)T^{2} \)
79 \( 1 + (0.739 + 0.853i)T + (-11.2 + 78.1i)T^{2} \)
83 \( 1 + (-3.84 + 1.12i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (1.35 + 9.45i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + (-1.07 - 0.315i)T + (81.6 + 52.4i)T^{2} \)
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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

−11.62567100371440174861387807682, −10.24509069546799548555825378990, −10.07092158645733800232135946438, −9.001789712037036467026113700252, −7.74926847939548934092045396243, −7.41570270144840824480845108338, −5.79845070873265628858532083305, −4.86807065736210396449489619543, −3.17602857170913053762281366897, −2.22609868106672032291541574105, 0.12071622720488404338253531590, 2.29650531618561966296893804459, 3.80663306102771757322678215173, 5.23787438351380780154578569683, 6.20297558095337865468293836849, 7.30114001831282192648500147036, 7.996966495088000861993468237533, 8.979828724429986883897347628617, 10.05244968894610364723557869471, 10.55910933640126210446271732633

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