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Properties

Label 2-10e2-20.3-c1-0-0
Degree $2$
Conductor $100$
Sign $-0.920 - 0.390i$
Analytic cond. $0.798504$
Root an. cond. $0.893590$
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  + (0.178 + 1.40i)2-s + (−1.58 + 1.58i)3-s + (−1.93 + 0.5i)4-s + (−2.50 − 1.93i)6-s + (−1.04 − 2.62i)8-s − 2.00i·9-s + 3.87i·11-s + (2.27 − 3.85i)12-s + (2.44 + 2.44i)13-s + (3.50 − 1.93i)16-s + (−1.22 + 1.22i)17-s + (2.80 − 0.356i)18-s + 3.87·19-s + (−5.43 + 0.690i)22-s + (3.16 − 3.16i)23-s + (5.80 + 2.5i)24-s + ⋯
L(s)  = 1  + (0.126 + 0.992i)2-s + (−0.912 + 0.912i)3-s + (−0.968 + 0.250i)4-s + (−1.02 − 0.790i)6-s + (−0.370 − 0.929i)8-s − 0.666i·9-s + 1.16i·11-s + (0.655 − 1.11i)12-s + (0.679 + 0.679i)13-s + (0.875 − 0.484i)16-s + (−0.297 + 0.297i)17-s + (0.661 − 0.0840i)18-s + 0.888·19-s + (−1.15 + 0.147i)22-s + (0.659 − 0.659i)23-s + (1.18 + 0.510i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $-0.920 - 0.390i$
Analytic conductor: \(0.798504\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :1/2),\ -0.920 - 0.390i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.143458 + 0.704857i\)
\(L(\frac12)\) \(\approx\) \(0.143458 + 0.704857i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.178 - 1.40i)T \)
5 \( 1 \)
good3 \( 1 + (1.58 - 1.58i)T - 3iT^{2} \)
7 \( 1 + 7iT^{2} \)
11 \( 1 - 3.87iT - 11T^{2} \)
13 \( 1 + (-2.44 - 2.44i)T + 13iT^{2} \)
17 \( 1 + (1.22 - 1.22i)T - 17iT^{2} \)
19 \( 1 - 3.87T + 19T^{2} \)
23 \( 1 + (-3.16 + 3.16i)T - 23iT^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 + 7.74iT - 31T^{2} \)
37 \( 1 + (-4.89 + 4.89i)T - 37iT^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (-3.16 - 3.16i)T + 47iT^{2} \)
53 \( 1 + (-2.44 - 2.44i)T + 53iT^{2} \)
59 \( 1 + 7.74T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + (-4.74 - 4.74i)T + 67iT^{2} \)
71 \( 1 - 7.74iT - 71T^{2} \)
73 \( 1 + (3.67 + 3.67i)T + 73iT^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 + (-1.58 + 1.58i)T - 83iT^{2} \)
89 \( 1 + 9iT - 89T^{2} \)
97 \( 1 + (-4.89 + 4.89i)T - 97iT^{2} \)
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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

−14.67542194624050721636510537484, −13.43395109437383051689948818145, −12.29116457259542737476230269727, −11.07773724997059934153128582671, −9.917084702518692040968459234247, −8.996256799795590422668886638854, −7.42766243236485520427919435254, −6.22371850199807023189197318189, −5.03408410897741448179560796386, −4.09557482877572237590136572092, 1.01644229967772285059616179637, 3.25918007032581744874915288191, 5.25136557700307347266929608206, 6.25909002686103229703502461716, 7.914333125891463901939862405195, 9.210298773541255059925592427787, 10.68772123168303758056115769258, 11.42259768772416056188912151481, 12.20385119259834384710951630867, 13.30794283828935799947615384050

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