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Properties

Label 2-10e2-20.3-c1-0-2
Degree $2$
Conductor $100$
Sign $0.648 - 0.761i$
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  + (1.40 + 0.178i)2-s + (−1.58 + 1.58i)3-s + (1.93 + 0.5i)4-s + (−2.50 + 1.93i)6-s + (2.62 + 1.04i)8-s − 2.00i·9-s − 3.87i·11-s + (−3.85 + 2.27i)12-s + (−2.44 − 2.44i)13-s + (3.50 + 1.93i)16-s + (1.22 − 1.22i)17-s + (0.356 − 2.80i)18-s − 3.87·19-s + (0.690 − 5.43i)22-s + (3.16 − 3.16i)23-s + (−5.80 + 2.5i)24-s + ⋯
L(s)  = 1  + (0.992 + 0.126i)2-s + (−0.912 + 0.912i)3-s + (0.968 + 0.250i)4-s + (−1.02 + 0.790i)6-s + (0.929 + 0.370i)8-s − 0.666i·9-s − 1.16i·11-s + (−1.11 + 0.655i)12-s + (−0.679 − 0.679i)13-s + (0.875 + 0.484i)16-s + (0.297 − 0.297i)17-s + (0.0840 − 0.661i)18-s − 0.888·19-s + (0.147 − 1.15i)22-s + (0.659 − 0.659i)23-s + (−1.18 + 0.510i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 - 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.648 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.648 - 0.761i$
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: \((2,\ 100,\ (\ :1/2),\ 0.648 - 0.761i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22206 + 0.564330i\)
\(L(\frac12)\) \(\approx\) \(1.22206 + 0.564330i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.178i)T \)
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

−14.12538766811977793958047565819, −12.90751664950564195051737980881, −11.91943695937272630276416403644, −10.88565363127355305778032927976, −10.30656347996286258021353863852, −8.468388862261355010666409935266, −6.82202183538480614715325653135, −5.56958560847115564699823060044, −4.78057691041202483230307347317, −3.23567872089790754359799627676, 2.01134324415544411997996953534, 4.31083376144405042131780703114, 5.64232965579589898521278094368, 6.74753573547781736613773898989, 7.54043519595380352667741144224, 9.716838053793883685238134111356, 11.03075100353443684471042307171, 11.97273885632799541813311421034, 12.56632046427516251374725692452, 13.43960048065297380360918093059

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