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Properties

Label 2-414-23.9-c1-0-9
Degree $2$
Conductor $414$
Sign $-0.0767 + 0.997i$
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  + (0.959 − 0.281i)2-s + (0.841 − 0.540i)4-s + (−0.455 − 3.17i)5-s + (0.628 − 1.37i)7-s + (0.654 − 0.755i)8-s + (−1.33 − 2.91i)10-s + (−6.27 − 1.84i)11-s + (1.19 + 2.62i)13-s + (0.215 − 1.49i)14-s + (0.415 − 0.909i)16-s + (1.00 + 0.646i)17-s + (0.467 − 0.300i)19-s + (−2.09 − 2.42i)20-s − 6.53·22-s + (4.66 − 1.10i)23-s + ⋯
L(s)  = 1  + (0.678 − 0.199i)2-s + (0.420 − 0.270i)4-s + (−0.203 − 1.41i)5-s + (0.237 − 0.520i)7-s + (0.231 − 0.267i)8-s + (−0.420 − 0.921i)10-s + (−1.89 − 0.555i)11-s + (0.332 + 0.728i)13-s + (0.0575 − 0.400i)14-s + (0.103 − 0.227i)16-s + (0.244 + 0.156i)17-s + (0.107 − 0.0689i)19-s + (−0.469 − 0.541i)20-s − 1.39·22-s + (0.973 − 0.229i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0767 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0767 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $-0.0767 + 0.997i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ -0.0767 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24744 - 1.34716i\)
\(L(\frac12)\) \(\approx\) \(1.24744 - 1.34716i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.959 + 0.281i)T \)
3 \( 1 \)
23 \( 1 + (-4.66 + 1.10i)T \)
good5 \( 1 + (0.455 + 3.17i)T + (-4.79 + 1.40i)T^{2} \)
7 \( 1 + (-0.628 + 1.37i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (6.27 + 1.84i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (-1.19 - 2.62i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (-1.00 - 0.646i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (-0.467 + 0.300i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (-7.29 - 4.68i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (-4.41 + 5.09i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (-0.206 + 1.43i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (0.262 + 1.82i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (1.42 + 1.64i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 - 11.4T + 47T^{2} \)
53 \( 1 + (5.01 - 10.9i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (1.64 + 3.60i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (5.91 - 6.82i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (-7.35 + 2.16i)T + (56.3 - 36.2i)T^{2} \)
71 \( 1 + (9.07 - 2.66i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (0.527 - 0.339i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (5.72 + 12.5i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (2.21 - 15.4i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (3.67 + 4.24i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (-1.03 - 7.17i)T + (-93.0 + 27.3i)T^{2} \)
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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

−10.99329684472129661210071255494, −10.36712462184438533390866120985, −9.046742737625852699269070135077, −8.237273534847083718926050465328, −7.32675006498520296035346321339, −5.87356419007906478371810833453, −4.97041830999194134613605213001, −4.29174043319473239387320488681, −2.78710409983951267934246337446, −1.01340778578659845265037417052, 2.57625609647936532558051501176, 3.13107111612767804677425540605, 4.78569909216194084452200471921, 5.66825060102878216017224594430, 6.76051428220493424589877598705, 7.61581048558958690808174282284, 8.354946697295230196716003050093, 10.08472447421270814683088646020, 10.57918752661815196088471645323, 11.44232928290015910746454186541

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