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Properties

Label 2-10e2-4.3-c10-0-44
Degree $2$
Conductor $100$
Sign $0.902 - 0.430i$
Analytic cond. $63.5357$
Root an. cond. $7.97093$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.05 − 31.2i)2-s + 442. i·3-s + (−924. + 440. i)4-s + (1.38e4 − 3.12e3i)6-s − 2.45e3i·7-s + (2.02e4 + 2.57e4i)8-s − 1.36e5·9-s + 6.73e4i·11-s + (−1.94e5 − 4.08e5i)12-s + 7.23e5·13-s + (−7.66e4 + 1.73e4i)14-s + (6.60e5 − 8.14e5i)16-s + 8.17e5·17-s + (9.64e5 + 4.26e6i)18-s − 2.29e6i·19-s + ⋯
L(s)  = 1  + (−0.220 − 0.975i)2-s + 1.82i·3-s + (−0.902 + 0.430i)4-s + (1.77 − 0.401i)6-s − 0.146i·7-s + (0.618 + 0.785i)8-s − 2.31·9-s + 0.418i·11-s + (−0.783 − 1.64i)12-s + 1.94·13-s + (−0.142 + 0.0322i)14-s + (0.629 − 0.776i)16-s + 0.575·17-s + (0.510 + 2.25i)18-s − 0.927i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.902 - 0.430i$
Analytic conductor: \(63.5357\)
Root analytic conductor: \(7.97093\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :5),\ 0.902 - 0.430i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.73172 + 0.391483i\)
\(L(\frac12)\) \(\approx\) \(1.73172 + 0.391483i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (7.05 + 31.2i)T \)
5 \( 1 \)
good3 \( 1 - 442. iT - 5.90e4T^{2} \)
7 \( 1 + 2.45e3iT - 2.82e8T^{2} \)
11 \( 1 - 6.73e4iT - 2.59e10T^{2} \)
13 \( 1 - 7.23e5T + 1.37e11T^{2} \)
17 \( 1 - 8.17e5T + 2.01e12T^{2} \)
19 \( 1 + 2.29e6iT - 6.13e12T^{2} \)
23 \( 1 + 6.87e6iT - 4.14e13T^{2} \)
29 \( 1 + 9.72e6T + 4.20e14T^{2} \)
31 \( 1 + 3.08e7iT - 8.19e14T^{2} \)
37 \( 1 - 1.03e8T + 4.80e15T^{2} \)
41 \( 1 - 1.28e8T + 1.34e16T^{2} \)
43 \( 1 - 7.29e7iT - 2.16e16T^{2} \)
47 \( 1 - 1.33e8iT - 5.25e16T^{2} \)
53 \( 1 + 2.17e8T + 1.74e17T^{2} \)
59 \( 1 + 9.48e8iT - 5.11e17T^{2} \)
61 \( 1 + 9.30e7T + 7.13e17T^{2} \)
67 \( 1 - 1.87e9iT - 1.82e18T^{2} \)
71 \( 1 - 5.98e8iT - 3.25e18T^{2} \)
73 \( 1 - 3.66e8T + 4.29e18T^{2} \)
79 \( 1 - 2.46e9iT - 9.46e18T^{2} \)
83 \( 1 - 4.63e9iT - 1.55e19T^{2} \)
89 \( 1 - 2.24e9T + 3.11e19T^{2} \)
97 \( 1 - 6.10e9T + 7.37e19T^{2} \)
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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

−11.27652319780206186874519720446, −10.96697986600118506857365474621, −9.873656199408928491497420965414, −9.133516240964342737433827933835, −8.181281895700288477009198994385, −5.83686838425524449366522756323, −4.50226197855366835188520976955, −3.83094782777632251315256827644, −2.68530751493676898774582530685, −0.76226838581154158795444291790, 0.799488008092810647402495931289, 1.54918593462244731881890130079, 3.50434441633964460555551357361, 5.73992413313085121127044475038, 6.17792543956905694185932980870, 7.42020566878222453979707007617, 8.167108768886235066673430653103, 9.042637185257249119657636857272, 10.84988313991851786544955097490, 12.07585051043600119670052036667

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