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Properties

Label 2-10e2-20.7-c9-0-12
Degree $2$
Conductor $100$
Sign $-0.389 - 0.921i$
Analytic cond. $51.5035$
Root an. cond. $7.17659$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (21.5 + 6.87i)2-s + (−110. − 110. i)3-s + (417. + 296. i)4-s + (−1.62e3 − 3.13e3i)6-s + (−1.21e3 + 1.21e3i)7-s + (6.96e3 + 9.25e3i)8-s + 4.67e3i·9-s + 3.90e4i·11-s + (−1.33e4 − 7.87e4i)12-s + (5.35e4 − 5.35e4i)13-s + (−3.45e4 + 1.78e4i)14-s + (8.64e4 + 2.47e5i)16-s + (−3.81e5 − 3.81e5i)17-s + (−3.21e4 + 1.00e5i)18-s − 5.75e5·19-s + ⋯
L(s)  = 1  + (0.952 + 0.303i)2-s + (−0.786 − 0.786i)3-s + (0.815 + 0.578i)4-s + (−0.510 − 0.988i)6-s + (−0.191 + 0.191i)7-s + (0.601 + 0.799i)8-s + 0.237i·9-s + 0.803i·11-s + (−0.186 − 1.09i)12-s + (0.519 − 0.519i)13-s + (−0.240 + 0.124i)14-s + (0.329 + 0.944i)16-s + (−1.10 − 1.10i)17-s + (−0.0721 + 0.226i)18-s − 1.01·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.389 - 0.921i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.389 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $-0.389 - 0.921i$
Analytic conductor: \(51.5035\)
Root analytic conductor: \(7.17659\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :9/2),\ -0.389 - 0.921i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.870953 + 1.31375i\)
\(L(\frac12)\) \(\approx\) \(0.870953 + 1.31375i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-21.5 - 6.87i)T \)
5 \( 1 \)
good3 \( 1 + (110. + 110. i)T + 1.96e4iT^{2} \)
7 \( 1 + (1.21e3 - 1.21e3i)T - 4.03e7iT^{2} \)
11 \( 1 - 3.90e4iT - 2.35e9T^{2} \)
13 \( 1 + (-5.35e4 + 5.35e4i)T - 1.06e10iT^{2} \)
17 \( 1 + (3.81e5 + 3.81e5i)T + 1.18e11iT^{2} \)
19 \( 1 + 5.75e5T + 3.22e11T^{2} \)
23 \( 1 + (-1.06e6 - 1.06e6i)T + 1.80e12iT^{2} \)
29 \( 1 - 3.51e6iT - 1.45e13T^{2} \)
31 \( 1 - 4.45e6iT - 2.64e13T^{2} \)
37 \( 1 + (5.44e6 + 5.44e6i)T + 1.29e14iT^{2} \)
41 \( 1 - 2.53e6T + 3.27e14T^{2} \)
43 \( 1 + (-2.19e7 - 2.19e7i)T + 5.02e14iT^{2} \)
47 \( 1 + (1.86e7 - 1.86e7i)T - 1.11e15iT^{2} \)
53 \( 1 + (6.92e7 - 6.92e7i)T - 3.29e15iT^{2} \)
59 \( 1 - 1.64e8T + 8.66e15T^{2} \)
61 \( 1 + 1.57e6T + 1.16e16T^{2} \)
67 \( 1 + (2.14e8 - 2.14e8i)T - 2.72e16iT^{2} \)
71 \( 1 + 5.04e7iT - 4.58e16T^{2} \)
73 \( 1 + (4.03e7 - 4.03e7i)T - 5.88e16iT^{2} \)
79 \( 1 + 3.84e8T + 1.19e17T^{2} \)
83 \( 1 + (5.30e8 + 5.30e8i)T + 1.86e17iT^{2} \)
89 \( 1 - 4.15e8iT - 3.50e17T^{2} \)
97 \( 1 + (8.68e7 + 8.68e7i)T + 7.60e17iT^{2} \)
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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

−12.70017644865481260348547372739, −11.59655819683766835997180227440, −10.79792601996416527196401048799, −8.985339048763564330766378107898, −7.39091898998041418505867234547, −6.68703115856680950605824922898, −5.66494020434320341549574221349, −4.51353865123338151427135918544, −2.88833747733857562711402414852, −1.42968672241222507506762570203, 0.31554222912784009580542663713, 2.10405396371529372769750555189, 3.79177790328242831605141750317, 4.57241846687182493852302266754, 5.86243044633715678916694571132, 6.63091179792976656968252032875, 8.549452831395181397583197788459, 10.11181217521089454377166773193, 10.93542667095449169319853931397, 11.47231736382770852034050564253

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