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Properties

Label 2-10e2-20.3-c3-0-9
Degree $2$
Conductor $100$
Sign $0.999 + 0.0300i$
Analytic cond. $5.90019$
Root an. cond. $2.42903$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.70 − 0.813i)2-s + (2.61 − 2.61i)3-s + (6.67 + 4.40i)4-s + (−9.22 + 4.96i)6-s + (17.7 + 17.7i)7-s + (−14.4 − 17.3i)8-s + 13.2i·9-s + 7.37i·11-s + (29.0 − 5.93i)12-s + (2.68 + 2.68i)13-s + (−33.6 − 62.6i)14-s + (25.1 + 58.8i)16-s + (20.2 − 20.2i)17-s + (10.8 − 36.0i)18-s + 135.·19-s + ⋯
L(s)  = 1  + (−0.957 − 0.287i)2-s + (0.503 − 0.503i)3-s + (0.834 + 0.551i)4-s + (−0.627 + 0.337i)6-s + (0.959 + 0.959i)7-s + (−0.640 − 0.767i)8-s + 0.492i·9-s + 0.202i·11-s + (0.698 − 0.142i)12-s + (0.0572 + 0.0572i)13-s + (−0.643 − 1.19i)14-s + (0.392 + 0.919i)16-s + (0.288 − 0.288i)17-s + (0.141 − 0.471i)18-s + 1.63·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0300i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.999 + 0.0300i$
Analytic conductor: \(5.90019\)
Root analytic conductor: \(2.42903\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :3/2),\ 0.999 + 0.0300i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.33949 - 0.0201430i\)
\(L(\frac12)\) \(\approx\) \(1.33949 - 0.0201430i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.70 + 0.813i)T \)
5 \( 1 \)
good3 \( 1 + (-2.61 + 2.61i)T - 27iT^{2} \)
7 \( 1 + (-17.7 - 17.7i)T + 343iT^{2} \)
11 \( 1 - 7.37iT - 1.33e3T^{2} \)
13 \( 1 + (-2.68 - 2.68i)T + 2.19e3iT^{2} \)
17 \( 1 + (-20.2 + 20.2i)T - 4.91e3iT^{2} \)
19 \( 1 - 135.T + 6.85e3T^{2} \)
23 \( 1 + (-71.0 + 71.0i)T - 1.21e4iT^{2} \)
29 \( 1 - 34.2iT - 2.43e4T^{2} \)
31 \( 1 + 187. iT - 2.97e4T^{2} \)
37 \( 1 + (250. - 250. i)T - 5.06e4iT^{2} \)
41 \( 1 + 211.T + 6.89e4T^{2} \)
43 \( 1 + (46.7 - 46.7i)T - 7.95e4iT^{2} \)
47 \( 1 + (189. + 189. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-74.5 - 74.5i)T + 1.48e5iT^{2} \)
59 \( 1 - 101.T + 2.05e5T^{2} \)
61 \( 1 - 232.T + 2.26e5T^{2} \)
67 \( 1 + (-34.7 - 34.7i)T + 3.00e5iT^{2} \)
71 \( 1 - 614. iT - 3.57e5T^{2} \)
73 \( 1 + (-37.4 - 37.4i)T + 3.89e5iT^{2} \)
79 \( 1 + 1.00e3T + 4.93e5T^{2} \)
83 \( 1 + (-423. + 423. i)T - 5.71e5iT^{2} \)
89 \( 1 + 1.04e3iT - 7.04e5T^{2} \)
97 \( 1 + (-536. + 536. i)T - 9.12e5iT^{2} \)
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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

−13.27027946523404216081744704859, −12.03016793554756856039088127545, −11.35647398766271569430768123702, −10.02151206951085619312112079512, −8.804811726395156738323918685245, −8.053654219990090887067024729781, −7.06364559222069221740173718169, −5.24906076288877976356633923764, −2.85281532681978631350675495502, −1.57815895977376737502752037466, 1.19020674666937910016036605111, 3.42895423101713222590083666024, 5.24472151034150427183457957007, 6.97014464409463253919778029904, 7.955944534775914270946609110755, 9.024298154338328098584608482924, 10.01346198034655310387401772297, 10.95875645279641168272941134065, 11.98348971679736117485392238752, 13.85432991157261767643800189637

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