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Properties

Label 2-10e2-20.3-c3-0-12
Degree $2$
Conductor $100$
Sign $0.806 + 0.591i$
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  + (−2.77 + 0.540i)2-s + (−2.23 + 2.23i)3-s + (7.41 − 3.00i)4-s + (5 − 7.41i)6-s + (−11.1 − 11.1i)7-s + (−18.9 + 12.3i)8-s + 17i·9-s − 59.3i·11-s + (−9.87 + 23.2i)12-s + (26.5 + 26.5i)13-s + (37.0 + 25.0i)14-s + (46.0 − 44.4i)16-s + (79.5 − 79.5i)17-s + (−9.18 − 47.1i)18-s + 59.3·19-s + ⋯
L(s)  = 1  + (−0.981 + 0.191i)2-s + (−0.430 + 0.430i)3-s + (0.927 − 0.375i)4-s + (0.340 − 0.504i)6-s + (−0.603 − 0.603i)7-s + (−0.838 + 0.545i)8-s + 0.629i·9-s − 1.62i·11-s + (−0.237 + 0.560i)12-s + (0.566 + 0.566i)13-s + (0.707 + 0.477i)14-s + (0.718 − 0.695i)16-s + (1.13 − 1.13i)17-s + (−0.120 − 0.618i)18-s + 0.716·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.806 + 0.591i$
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: \((2,\ 100,\ (\ :3/2),\ 0.806 + 0.591i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.718098 - 0.235114i\)
\(L(\frac12)\) \(\approx\) \(0.718098 - 0.235114i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.77 - 0.540i)T \)
5 \( 1 \)
good3 \( 1 + (2.23 - 2.23i)T - 27iT^{2} \)
7 \( 1 + (11.1 + 11.1i)T + 343iT^{2} \)
11 \( 1 + 59.3iT - 1.33e3T^{2} \)
13 \( 1 + (-26.5 - 26.5i)T + 2.19e3iT^{2} \)
17 \( 1 + (-79.5 + 79.5i)T - 4.91e3iT^{2} \)
19 \( 1 - 59.3T + 6.85e3T^{2} \)
23 \( 1 + (-105. + 105. i)T - 1.21e4iT^{2} \)
29 \( 1 + 46iT - 2.43e4T^{2} \)
31 \( 1 - 118. iT - 2.97e4T^{2} \)
37 \( 1 + (-212. + 212. i)T - 5.06e4iT^{2} \)
41 \( 1 + 188T + 6.89e4T^{2} \)
43 \( 1 + (122. - 122. i)T - 7.95e4iT^{2} \)
47 \( 1 + (194. + 194. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-26.5 - 26.5i)T + 1.48e5iT^{2} \)
59 \( 1 + 415.T + 2.05e5T^{2} \)
61 \( 1 - 72T + 2.26e5T^{2} \)
67 \( 1 + (-597. - 597. i)T + 3.00e5iT^{2} \)
71 \( 1 + 711. iT - 3.57e5T^{2} \)
73 \( 1 + (504. + 504. i)T + 3.89e5iT^{2} \)
79 \( 1 - 830.T + 4.93e5T^{2} \)
83 \( 1 + (221. - 221. i)T - 5.71e5iT^{2} \)
89 \( 1 + 726iT - 7.04e5T^{2} \)
97 \( 1 + (451. - 451. i)T - 9.12e5iT^{2} \)
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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

−13.44488997085868636459175825600, −11.71083265418288963425641229064, −10.96292420707462651456001918310, −10.08668897908729356754738324695, −8.991289162835987866022561678659, −7.79238491447674980597615782639, −6.53313515271554001953596039415, −5.32486688392461788477060959942, −3.19206730455105442158998667100, −0.70134386012158524162663998284, 1.36807359082998280261698352363, 3.29479912885895592458455470391, 5.75660793897795037256799758374, 6.84354111293640668181400905097, 7.928403355045924676597937422226, 9.366387862459576782636661885568, 10.02367181904121533404651976394, 11.42763409894115885876837693245, 12.37668601852294059341463910878, 12.89335067762260796244592815028

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