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Properties

Label 2-40e2-80.27-c1-0-23
Degree $2$
Conductor $1600$
Sign $0.700 + 0.713i$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
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  + 1.86·3-s + (−0.719 − 0.719i)7-s + 0.487·9-s + (0.805 − 0.805i)11-s − 5.90i·13-s + (5.17 + 5.17i)17-s + (1.16 − 1.16i)19-s + (−1.34 − 1.34i)21-s + (2.30 − 2.30i)23-s − 4.69·27-s + (3.71 + 3.71i)29-s − 9.82i·31-s + (1.50 − 1.50i)33-s + 1.71i·37-s − 11.0i·39-s + ⋯
L(s)  = 1  + 1.07·3-s + (−0.272 − 0.272i)7-s + 0.162·9-s + (0.242 − 0.242i)11-s − 1.63i·13-s + (1.25 + 1.25i)17-s + (0.266 − 0.266i)19-s + (−0.293 − 0.293i)21-s + (0.479 − 0.479i)23-s − 0.902·27-s + (0.690 + 0.690i)29-s − 1.76i·31-s + (0.261 − 0.261i)33-s + 0.282i·37-s − 1.76i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.700 + 0.713i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.700 + 0.713i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.395469465\)
\(L(\frac12)\) \(\approx\) \(2.395469465\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 - 1.86T + 3T^{2} \)
7 \( 1 + (0.719 + 0.719i)T + 7iT^{2} \)
11 \( 1 + (-0.805 + 0.805i)T - 11iT^{2} \)
13 \( 1 + 5.90iT - 13T^{2} \)
17 \( 1 + (-5.17 - 5.17i)T + 17iT^{2} \)
19 \( 1 + (-1.16 + 1.16i)T - 19iT^{2} \)
23 \( 1 + (-2.30 + 2.30i)T - 23iT^{2} \)
29 \( 1 + (-3.71 - 3.71i)T + 29iT^{2} \)
31 \( 1 + 9.82iT - 31T^{2} \)
37 \( 1 - 1.71iT - 37T^{2} \)
41 \( 1 - 3.93iT - 41T^{2} \)
43 \( 1 + 8.82iT - 43T^{2} \)
47 \( 1 + (-3.21 + 3.21i)T - 47iT^{2} \)
53 \( 1 - 8.60T + 53T^{2} \)
59 \( 1 + (5.24 + 5.24i)T + 59iT^{2} \)
61 \( 1 + (-1.59 + 1.59i)T - 61iT^{2} \)
67 \( 1 - 9.29iT - 67T^{2} \)
71 \( 1 + 9.33T + 71T^{2} \)
73 \( 1 + (-8.57 - 8.57i)T + 73iT^{2} \)
79 \( 1 - 1.70T + 79T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + 4.48T + 89T^{2} \)
97 \( 1 + (-4.46 - 4.46i)T + 97iT^{2} \)
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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

−9.211800712292709216673119222676, −8.330545043437840205908045626595, −8.011079556129665386594795072540, −7.10361221662880811271060987392, −5.98100451534643093528055342413, −5.29778947400624107914344367062, −3.86213274053516804347347780340, −3.30436353360331428110934489333, −2.41447071672657171705137308863, −0.884536894020516246258803948595, 1.43219756045359398298001097133, 2.63268441994604723955494487224, 3.33259709566354840007703580415, 4.36070179633433265439810333082, 5.34471137801760573207954384006, 6.42083283466771678792839068026, 7.27179794105350577127569670629, 7.908754862719838957828940804712, 9.013675029105742644133186687560, 9.228394432962085096340402375891

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