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Properties

Label 2-10e2-20.19-c8-0-22
Degree $2$
Conductor $100$
Sign $0.897 - 0.441i$
Analytic cond. $40.7378$
Root an. cond. $6.38262$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−11.2 − 11.3i)2-s − 137.·3-s + (−1.63 + 255. i)4-s + (1.54e3 + 1.55e3i)6-s + 3.94e3·7-s + (2.92e3 − 2.86e3i)8-s + 1.22e4·9-s + 1.70e4i·11-s + (224. − 3.51e4i)12-s + 1.09e4i·13-s + (−4.44e4 − 4.47e4i)14-s + (−6.55e4 − 837. i)16-s − 1.01e5i·17-s + (−1.38e5 − 1.39e5i)18-s + 9.34e4i·19-s + ⋯
L(s)  = 1  + (−0.704 − 0.709i)2-s − 1.69·3-s + (−0.00639 + 0.999i)4-s + (1.19 + 1.20i)6-s + 1.64·7-s + (0.713 − 0.700i)8-s + 1.87·9-s + 1.16i·11-s + (0.0108 − 1.69i)12-s + 0.382i·13-s + (−1.15 − 1.16i)14-s + (−0.999 − 0.0127i)16-s − 1.21i·17-s + (−1.32 − 1.32i)18-s + 0.716i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.897 - 0.441i$
Analytic conductor: \(40.7378\)
Root analytic conductor: \(6.38262\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :4),\ 0.897 - 0.441i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.799291 + 0.185992i\)
\(L(\frac12)\) \(\approx\) \(0.799291 + 0.185992i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (11.2 + 11.3i)T \)
5 \( 1 \)
good3 \( 1 + 137.T + 6.56e3T^{2} \)
7 \( 1 - 3.94e3T + 5.76e6T^{2} \)
11 \( 1 - 1.70e4iT - 2.14e8T^{2} \)
13 \( 1 - 1.09e4iT - 8.15e8T^{2} \)
17 \( 1 + 1.01e5iT - 6.97e9T^{2} \)
19 \( 1 - 9.34e4iT - 1.69e10T^{2} \)
23 \( 1 - 1.47e5T + 7.83e10T^{2} \)
29 \( 1 + 4.16e4T + 5.00e11T^{2} \)
31 \( 1 - 1.38e5iT - 8.52e11T^{2} \)
37 \( 1 + 1.14e6iT - 3.51e12T^{2} \)
41 \( 1 - 3.83e6T + 7.98e12T^{2} \)
43 \( 1 + 3.18e6T + 1.16e13T^{2} \)
47 \( 1 + 3.51e6T + 2.38e13T^{2} \)
53 \( 1 - 5.66e6iT - 6.22e13T^{2} \)
59 \( 1 + 1.69e7iT - 1.46e14T^{2} \)
61 \( 1 + 5.16e6T + 1.91e14T^{2} \)
67 \( 1 - 1.05e7T + 4.06e14T^{2} \)
71 \( 1 - 1.85e7iT - 6.45e14T^{2} \)
73 \( 1 - 2.38e6iT - 8.06e14T^{2} \)
79 \( 1 - 4.42e7iT - 1.51e15T^{2} \)
83 \( 1 + 1.51e7T + 2.25e15T^{2} \)
89 \( 1 - 5.42e7T + 3.93e15T^{2} \)
97 \( 1 - 1.24e8iT - 7.83e15T^{2} \)
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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

−11.87647448559099223088076860852, −11.35494828012416369775901720600, −10.52803840072188335548378636147, −9.430268074616345615236091984975, −7.83125345116638583878776072914, −6.91306744521410270600909673964, −5.15097195146163025306928613573, −4.38809631376868514491900373936, −1.94379523221270752462759189803, −0.925750696268767733927700924731, 0.52785884686959291462012605496, 1.48470987979343733026640984090, 4.63665904354920672114512053623, 5.49520413270323458792997172263, 6.36228029313510528085443412380, 7.69650868369450884319354361529, 8.691174600232538279057002898078, 10.39850124165005646817960646396, 11.05368235315790723776402244647, 11.64186982931899542885301256733

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