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Properties

Label 2-10e2-20.19-c8-0-7
Degree $2$
Conductor $100$
Sign $-0.988 + 0.151i$
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  + (−14.2 − 7.35i)2-s − 110.·3-s + (147. + 208. i)4-s + (1.56e3 + 809. i)6-s − 3.54e3·7-s + (−567. − 4.05e3i)8-s + 5.57e3·9-s + 1.59e4i·11-s + (−1.62e4 − 2.30e4i)12-s + 2.41e4i·13-s + (5.03e4 + 2.60e4i)14-s + (−2.17e4 + 6.18e4i)16-s + 4.38e4i·17-s + (−7.92e4 − 4.09e4i)18-s − 5.09e4i·19-s + ⋯
L(s)  = 1  + (−0.888 − 0.459i)2-s − 1.36·3-s + (0.577 + 0.816i)4-s + (1.20 + 0.624i)6-s − 1.47·7-s + (−0.138 − 0.990i)8-s + 0.849·9-s + 1.09i·11-s + (−0.786 − 1.10i)12-s + 0.846i·13-s + (1.30 + 0.677i)14-s + (−0.332 + 0.943i)16-s + 0.524i·17-s + (−0.754 − 0.390i)18-s − 0.390i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $-0.988 + 0.151i$
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: \((2,\ 100,\ (\ :4),\ -0.988 + 0.151i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0135861 - 0.177791i\)
\(L(\frac12)\) \(\approx\) \(0.0135861 - 0.177791i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (14.2 + 7.35i)T \)
5 \( 1 \)
good3 \( 1 + 110.T + 6.56e3T^{2} \)
7 \( 1 + 3.54e3T + 5.76e6T^{2} \)
11 \( 1 - 1.59e4iT - 2.14e8T^{2} \)
13 \( 1 - 2.41e4iT - 8.15e8T^{2} \)
17 \( 1 - 4.38e4iT - 6.97e9T^{2} \)
19 \( 1 + 5.09e4iT - 1.69e10T^{2} \)
23 \( 1 - 2.70e5T + 7.83e10T^{2} \)
29 \( 1 - 1.32e6T + 5.00e11T^{2} \)
31 \( 1 - 1.18e6iT - 8.52e11T^{2} \)
37 \( 1 - 2.97e6iT - 3.51e12T^{2} \)
41 \( 1 + 4.92e6T + 7.98e12T^{2} \)
43 \( 1 + 2.86e6T + 1.16e13T^{2} \)
47 \( 1 - 1.26e4T + 2.38e13T^{2} \)
53 \( 1 - 5.50e6iT - 6.22e13T^{2} \)
59 \( 1 - 6.68e6iT - 1.46e14T^{2} \)
61 \( 1 + 2.50e6T + 1.91e14T^{2} \)
67 \( 1 + 3.86e6T + 4.06e14T^{2} \)
71 \( 1 - 3.21e7iT - 6.45e14T^{2} \)
73 \( 1 + 1.91e7iT - 8.06e14T^{2} \)
79 \( 1 + 5.33e7iT - 1.51e15T^{2} \)
83 \( 1 - 6.22e6T + 2.25e15T^{2} \)
89 \( 1 + 3.83e7T + 3.93e15T^{2} \)
97 \( 1 - 7.87e7iT - 7.83e15T^{2} \)
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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

−12.31833915822376136431064783610, −11.82817938390597499670248240543, −10.50359452247774739500886550471, −9.932597993828760778092761810745, −8.750531668511327699729015967888, −6.81712869538311950284256877788, −6.57953421173859206933713167614, −4.70882355736808016726850556217, −3.03536753069402348842880835756, −1.27165026338010520016112463972, 0.13930861565850514470268059135, 0.73096318974126168504941832081, 3.03579330647221819237451727096, 5.29522770702243117895708478907, 6.13992484600587489985946198381, 6.89801005893074112433802515320, 8.426131503717316764610553254501, 9.720055571863180729240450472582, 10.54112266663573736481302748855, 11.44364168137612287815905349938

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