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Properties

Label 2-10e2-20.19-c10-0-45
Degree $2$
Conductor $100$
Sign $0.403 - 0.914i$
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  + (31.9 − 0.770i)2-s + 80.5·3-s + (1.02e3 − 49.2i)4-s + (2.57e3 − 62.0i)6-s + 345.·7-s + (3.26e4 − 2.36e3i)8-s − 5.25e4·9-s + 1.67e5i·11-s + (8.23e4 − 3.97e3i)12-s − 9.56e4i·13-s + (1.10e4 − 265. i)14-s + (1.04e6 − 1.00e5i)16-s + 2.26e6i·17-s + (−1.68e6 + 4.04e4i)18-s + 1.35e6i·19-s + ⋯
L(s)  = 1  + (0.999 − 0.0240i)2-s + 0.331·3-s + (0.998 − 0.0481i)4-s + (0.331 − 0.00798i)6-s + 0.0205·7-s + (0.997 − 0.0721i)8-s − 0.890·9-s + 1.03i·11-s + (0.331 − 0.0159i)12-s − 0.257i·13-s + (0.0205 − 0.000494i)14-s + (0.995 − 0.0961i)16-s + 1.59i·17-s + (−0.889 + 0.0214i)18-s + 0.547i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 - 0.914i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.403 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.403 - 0.914i$
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: \((2,\ 100,\ (\ :5),\ 0.403 - 0.914i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(3.59877 + 2.34572i\)
\(L(\frac12)\) \(\approx\) \(3.59877 + 2.34572i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-31.9 + 0.770i)T \)
5 \( 1 \)
good3 \( 1 - 80.5T + 5.90e4T^{2} \)
7 \( 1 - 345.T + 2.82e8T^{2} \)
11 \( 1 - 1.67e5iT - 2.59e10T^{2} \)
13 \( 1 + 9.56e4iT - 1.37e11T^{2} \)
17 \( 1 - 2.26e6iT - 2.01e12T^{2} \)
19 \( 1 - 1.35e6iT - 6.13e12T^{2} \)
23 \( 1 - 7.72e6T + 4.14e13T^{2} \)
29 \( 1 - 6.87e6T + 4.20e14T^{2} \)
31 \( 1 - 4.09e7iT - 8.19e14T^{2} \)
37 \( 1 + 4.60e7iT - 4.80e15T^{2} \)
41 \( 1 - 1.22e7T + 1.34e16T^{2} \)
43 \( 1 - 2.14e8T + 2.16e16T^{2} \)
47 \( 1 + 1.55e8T + 5.25e16T^{2} \)
53 \( 1 - 5.32e7iT - 1.74e17T^{2} \)
59 \( 1 - 9.05e8iT - 5.11e17T^{2} \)
61 \( 1 - 7.30e8T + 7.13e17T^{2} \)
67 \( 1 - 5.67e8T + 1.82e18T^{2} \)
71 \( 1 + 2.07e9iT - 3.25e18T^{2} \)
73 \( 1 - 3.04e9iT - 4.29e18T^{2} \)
79 \( 1 + 2.49e9iT - 9.46e18T^{2} \)
83 \( 1 + 6.00e9T + 1.55e19T^{2} \)
89 \( 1 + 5.14e9T + 3.11e19T^{2} \)
97 \( 1 - 6.01e9iT - 7.37e19T^{2} \)
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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

−12.36888423829728115740895911991, −11.13913741844216921076102942694, −10.20353243118649552220913854862, −8.659038846885650618165318335599, −7.49400758437753007128666320929, −6.26480662760242445418489430291, −5.15165887620275175483384689452, −3.87619044313260139341577572660, −2.72761933275598163764551307370, −1.50356258646434309924947057868, 0.68933679807374788018165068581, 2.51140984437334736847705233379, 3.28008116498770425582654686814, 4.78112901627936797573607204453, 5.83477450220764094685293530338, 7.02201302910026156196882547895, 8.264141491940642069113118416477, 9.479725090542569656755023623536, 11.21562687824863914955189776160, 11.48521596694160912344981734439

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