Properties

Label 2-10e2-20.19-c8-0-3
Degree $2$
Conductor $100$
Sign $-0.970 + 0.240i$
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  + (−12.4 + 10i)2-s + 99.9·3-s + (56 − 249. i)4-s + (−1.24e3 + 999. i)6-s − 1.39e3·7-s + (1.79e3 + 3.68e3i)8-s + 3.42e3·9-s + 1.84e4i·11-s + (5.59e3 − 2.49e4i)12-s − 5.47e3i·13-s + (1.74e4 − 1.39e4i)14-s + (−5.92e4 − 2.79e4i)16-s − 7.30e4i·17-s + (−4.27e4 + 3.42e4i)18-s − 1.94e4i·19-s + ⋯
L(s)  = 1  + (−0.780 + 0.625i)2-s + 1.23·3-s + (0.218 − 0.975i)4-s + (−0.962 + 0.770i)6-s − 0.582·7-s + (0.439 + 0.898i)8-s + 0.521·9-s + 1.26i·11-s + (0.269 − 1.20i)12-s − 0.191i·13-s + (0.454 − 0.364i)14-s + (−0.904 − 0.426i)16-s − 0.875i·17-s + (−0.407 + 0.326i)18-s − 0.149i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.240i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $-0.970 + 0.240i$
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: \((2,\ 100,\ (\ :4),\ -0.970 + 0.240i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0498911 - 0.408410i\)
\(L(\frac12)\) \(\approx\) \(0.0498911 - 0.408410i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (12.4 - 10i)T \)
5 \( 1 \)
good3 \( 1 - 99.9T + 6.56e3T^{2} \)
7 \( 1 + 1.39e3T + 5.76e6T^{2} \)
11 \( 1 - 1.84e4iT - 2.14e8T^{2} \)
13 \( 1 + 5.47e3iT - 8.15e8T^{2} \)
17 \( 1 + 7.30e4iT - 6.97e9T^{2} \)
19 \( 1 + 1.94e4iT - 1.69e10T^{2} \)
23 \( 1 + 2.37e5T + 7.83e10T^{2} \)
29 \( 1 - 1.28e5T + 5.00e11T^{2} \)
31 \( 1 - 6.79e4iT - 8.52e11T^{2} \)
37 \( 1 - 3.47e6iT - 3.51e12T^{2} \)
41 \( 1 - 2.14e6T + 7.98e12T^{2} \)
43 \( 1 + 5.92e6T + 1.16e13T^{2} \)
47 \( 1 + 7.62e6T + 2.38e13T^{2} \)
53 \( 1 - 8.24e5iT - 6.22e13T^{2} \)
59 \( 1 + 3.72e6iT - 1.46e14T^{2} \)
61 \( 1 + 1.47e7T + 1.91e14T^{2} \)
67 \( 1 + 1.52e7T + 4.06e14T^{2} \)
71 \( 1 - 1.19e6iT - 6.45e14T^{2} \)
73 \( 1 + 5.72e6iT - 8.06e14T^{2} \)
79 \( 1 - 3.59e7iT - 1.51e15T^{2} \)
83 \( 1 + 5.19e7T + 2.25e15T^{2} \)
89 \( 1 - 8.33e7T + 3.93e15T^{2} \)
97 \( 1 + 1.20e8iT - 7.83e15T^{2} \)
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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

−13.16166947568040183526165255050, −11.69546934999299916843132075727, −10.02562091201467466492821396596, −9.556219131572751438261583987327, −8.432349343146399770368394896751, −7.53243070171058453850542772627, −6.47655235128846110415665277431, −4.78786928166834650524077022273, −2.98436368857395528350323442163, −1.73889060699870421058925503631, 0.11772010957281604821386250032, 1.78492462244009581562624005694, 3.04127856325343082494963667463, 3.82387419826093922081033033851, 6.23622569131502136025603076084, 7.77219108731046596782051733182, 8.546587045525649024636605088391, 9.360148888411605494588684155411, 10.41643349004049918393173404697, 11.55247674303943732400522970776

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