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Properties

Label 2-10e2-20.7-c3-0-5
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: $\chi_{100} (7, \cdot )$
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

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

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