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Properties

Label 2-10e2-5.4-c5-0-3
Degree $2$
Conductor $100$
Sign $0.447 - 0.894i$
Analytic cond. $16.0383$
Root an. cond. $4.00479$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 12i·3-s − 88i·7-s + 99·9-s + 540·11-s + 418i·13-s + 594i·17-s − 836·19-s + 1.05e3·21-s + 4.10e3i·23-s + 4.10e3i·27-s + 594·29-s + 4.25e3·31-s + 6.48e3i·33-s − 298i·37-s − 5.01e3·39-s + ⋯
L(s)  = 1  + 0.769i·3-s − 0.678i·7-s + 0.407·9-s + 1.34·11-s + 0.685i·13-s + 0.498i·17-s − 0.531·19-s + 0.522·21-s + 1.61i·23-s + 1.08i·27-s + 0.131·29-s + 0.795·31-s + 1.03i·33-s − 0.0357i·37-s − 0.528·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(16.0383\)
Root analytic conductor: \(4.00479\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :5/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.68475 + 1.04123i\)
\(L(\frac12)\) \(\approx\) \(1.68475 + 1.04123i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 - 12iT - 243T^{2} \)
7 \( 1 + 88iT - 1.68e4T^{2} \)
11 \( 1 - 540T + 1.61e5T^{2} \)
13 \( 1 - 418iT - 3.71e5T^{2} \)
17 \( 1 - 594iT - 1.41e6T^{2} \)
19 \( 1 + 836T + 2.47e6T^{2} \)
23 \( 1 - 4.10e3iT - 6.43e6T^{2} \)
29 \( 1 - 594T + 2.05e7T^{2} \)
31 \( 1 - 4.25e3T + 2.86e7T^{2} \)
37 \( 1 + 298iT - 6.93e7T^{2} \)
41 \( 1 - 1.72e4T + 1.15e8T^{2} \)
43 \( 1 - 1.21e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.29e3iT - 2.29e8T^{2} \)
53 \( 1 + 1.94e4iT - 4.18e8T^{2} \)
59 \( 1 - 7.66e3T + 7.14e8T^{2} \)
61 \( 1 + 3.47e4T + 8.44e8T^{2} \)
67 \( 1 - 2.18e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.68e4T + 1.80e9T^{2} \)
73 \( 1 + 6.75e4iT - 2.07e9T^{2} \)
79 \( 1 - 7.69e4T + 3.07e9T^{2} \)
83 \( 1 + 6.77e4iT - 3.93e9T^{2} \)
89 \( 1 + 2.97e4T + 5.58e9T^{2} \)
97 \( 1 + 1.22e5iT - 8.58e9T^{2} \)
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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

−13.20815479895680098001564353237, −11.89661695338397288241979451445, −10.89708806219967913385458770068, −9.822502134501987913785763487523, −9.026152304595149776645052717057, −7.44555903297766330524587557750, −6.27092091519014015084841797241, −4.50685572201736864303206024550, −3.71666978379047021140590370427, −1.40565260639191609453305475098, 0.913882871596800214533784300681, 2.46587976791816301424791845958, 4.33341920730405106686174517848, 6.03975676589006917803936088875, 6.97346389887750239379591014397, 8.274844268100986123986863333172, 9.335093011529123433969775117690, 10.65898617984728040008810232315, 12.05755404258012186985969404174, 12.51396853960862174236686145094

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