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Properties

Label 2-10e2-20.19-c10-0-47
Degree $2$
Conductor $100$
Sign $-0.951 + 0.308i$
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  + (−20.8 − 24.2i)2-s − 208.·3-s + (−153. + 1.01e3i)4-s + (4.35e3 + 5.06e3i)6-s − 1.75e4·7-s + (2.77e4 − 1.74e4i)8-s − 1.54e4·9-s − 2.65e5i·11-s + (3.20e4 − 2.11e5i)12-s + 6.47e5i·13-s + (3.66e5 + 4.25e5i)14-s + (−1.00e6 − 3.10e5i)16-s + 2.51e6i·17-s + (3.22e5 + 3.75e5i)18-s − 1.70e5i·19-s + ⋯
L(s)  = 1  + (−0.652 − 0.758i)2-s − 0.859·3-s + (−0.149 + 0.988i)4-s + (0.560 + 0.651i)6-s − 1.04·7-s + (0.847 − 0.531i)8-s − 0.261·9-s − 1.65i·11-s + (0.128 − 0.849i)12-s + 1.74i·13-s + (0.681 + 0.792i)14-s + (−0.955 − 0.296i)16-s + 1.77i·17-s + (0.170 + 0.198i)18-s − 0.0688i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $-0.951 + 0.308i$
Analytic conductor: \(63.5357\)
Root analytic conductor: \(7.97093\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :5),\ -0.951 + 0.308i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.0156780 - 0.0992480i\)
\(L(\frac12)\) \(\approx\) \(0.0156780 - 0.0992480i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (20.8 + 24.2i)T \)
5 \( 1 \)
good3 \( 1 + 208.T + 5.90e4T^{2} \)
7 \( 1 + 1.75e4T + 2.82e8T^{2} \)
11 \( 1 + 2.65e5iT - 2.59e10T^{2} \)
13 \( 1 - 6.47e5iT - 1.37e11T^{2} \)
17 \( 1 - 2.51e6iT - 2.01e12T^{2} \)
19 \( 1 + 1.70e5iT - 6.13e12T^{2} \)
23 \( 1 - 5.21e6T + 4.14e13T^{2} \)
29 \( 1 + 6.80e6T + 4.20e14T^{2} \)
31 \( 1 - 2.47e7iT - 8.19e14T^{2} \)
37 \( 1 + 9.23e6iT - 4.80e15T^{2} \)
41 \( 1 - 1.44e8T + 1.34e16T^{2} \)
43 \( 1 + 2.79e7T + 2.16e16T^{2} \)
47 \( 1 - 1.10e8T + 5.25e16T^{2} \)
53 \( 1 - 1.09e8iT - 1.74e17T^{2} \)
59 \( 1 - 6.65e8iT - 5.11e17T^{2} \)
61 \( 1 + 7.40e8T + 7.13e17T^{2} \)
67 \( 1 + 4.17e7T + 1.82e18T^{2} \)
71 \( 1 + 6.30e8iT - 3.25e18T^{2} \)
73 \( 1 - 9.52e8iT - 4.29e18T^{2} \)
79 \( 1 - 2.24e9iT - 9.46e18T^{2} \)
83 \( 1 + 6.38e9T + 1.55e19T^{2} \)
89 \( 1 - 1.29e9T + 3.11e19T^{2} \)
97 \( 1 + 1.43e10iT - 7.37e19T^{2} \)
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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

−11.20393652381320073086051892260, −10.63198947850785595388545363534, −9.237950029131086090790535954593, −8.502421450229253341532755520252, −6.77936038409013764240768671651, −5.90078477300172406405398140521, −4.06890723768001349674530244596, −2.95464723455804753763753152347, −1.29576765126569838835528415877, −0.05649516821811321837979156607, 0.73772941933681314947828211822, 2.71742817233799169414248547913, 4.84764446641227093314048212432, 5.71149636782158546563360200385, 6.84037908965263089548195792766, 7.69643353718041627380086483711, 9.310078406246055415323722417317, 10.00297335199419996230311534865, 11.04333774392127372021252435305, 12.33563905934230534923541070250

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