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Properties

Label 2-416-52.43-c2-0-23
Degree $2$
Conductor $416$
Sign $0.0347 + 0.999i$
Analytic cond. $11.3351$
Root an. cond. $3.36677$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.73 + 1.57i)3-s − 8.34i·5-s + (−1.62 − 2.80i)7-s + (0.476 + 0.825i)9-s + (−5.60 + 9.70i)11-s + (8.98 − 9.39i)13-s + (13.1 − 22.8i)15-s + (−8.55 − 14.8i)17-s + (−2.25 − 3.90i)19-s − 10.2i·21-s + (−17.1 − 9.90i)23-s − 44.6·25-s − 25.3i·27-s + (−7.78 + 13.4i)29-s + 41.8·31-s + ⋯
L(s)  = 1  + (0.910 + 0.525i)3-s − 1.66i·5-s + (−0.231 − 0.401i)7-s + (0.0529 + 0.0917i)9-s + (−0.509 + 0.882i)11-s + (0.691 − 0.722i)13-s + (0.877 − 1.52i)15-s + (−0.503 − 0.871i)17-s + (−0.118 − 0.205i)19-s − 0.487i·21-s + (−0.745 − 0.430i)23-s − 1.78·25-s − 0.940i·27-s + (−0.268 + 0.465i)29-s + 1.34·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0347 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0347 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.0347 + 0.999i$
Analytic conductor: \(11.3351\)
Root analytic conductor: \(3.36677\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1),\ 0.0347 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.36685 - 1.32018i\)
\(L(\frac12)\) \(\approx\) \(1.36685 - 1.32018i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (-8.98 + 9.39i)T \)
good3 \( 1 + (-2.73 - 1.57i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + 8.34iT - 25T^{2} \)
7 \( 1 + (1.62 + 2.80i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (5.60 - 9.70i)T + (-60.5 - 104. i)T^{2} \)
17 \( 1 + (8.55 + 14.8i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (2.25 + 3.90i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (17.1 + 9.90i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (7.78 - 13.4i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 - 41.8T + 961T^{2} \)
37 \( 1 + (41.1 + 23.7i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-49.3 - 28.4i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-39.8 + 22.9i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 - 36.3T + 2.20e3T^{2} \)
53 \( 1 - 14.8T + 2.80e3T^{2} \)
59 \( 1 + (-16.0 - 27.7i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-6.63 - 11.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (48.4 - 83.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-50.7 - 87.9i)T + (-2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 - 42.3iT - 5.32e3T^{2} \)
79 \( 1 + 147. iT - 6.24e3T^{2} \)
83 \( 1 + 61.0T + 6.88e3T^{2} \)
89 \( 1 + (-130. - 75.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-30.1 + 17.4i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.47098385934455708511854911665, −9.694506243162895809899681294419, −8.880867407000546366370541171743, −8.356118836471765421318820770953, −7.31349199540599072340484378823, −5.76654220737625169314327643265, −4.67250501148065907860356612944, −3.94306796887370092721394009339, −2.47595800515196371264410866311, −0.71750466231193281768053172372, 2.06562074979842144316656301115, 2.91736264079520958947305115380, 3.86008335994149429453337740146, 5.90643780068309310072126710979, 6.52511764325375696550117948242, 7.62482928200446077879467927765, 8.320534887986685651774066965889, 9.279452357573684064258953888782, 10.53764969169589166377363676797, 11.00569933418279682288086605497

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