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Properties

Label 2-10e2-4.3-c10-0-60
Degree $2$
Conductor $100$
Sign $0.669 - 0.743i$
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  + (13.0 + 29.2i)2-s + 448. i·3-s + (−685. + 760. i)4-s + (−1.31e4 + 5.84e3i)6-s − 1.93e4i·7-s + (−3.11e4 − 1.01e4i)8-s − 1.42e5·9-s − 2.07e5i·11-s + (−3.41e5 − 3.07e5i)12-s − 6.51e4·13-s + (5.65e5 − 2.51e5i)14-s + (−1.09e5 − 1.04e6i)16-s + 8.05e5·17-s + (−1.85e6 − 4.15e6i)18-s + 4.08e6i·19-s + ⋯
L(s)  = 1  + (0.406 + 0.913i)2-s + 1.84i·3-s + (−0.669 + 0.743i)4-s + (−1.68 + 0.751i)6-s − 1.15i·7-s + (−0.951 − 0.309i)8-s − 2.40·9-s − 1.28i·11-s + (−1.37 − 1.23i)12-s − 0.175·13-s + (1.05 − 0.467i)14-s + (−0.104 − 0.994i)16-s + 0.567·17-s + (−0.980 − 2.20i)18-s + 1.64i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.743i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.669 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.669 - 0.743i$
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: \((2,\ 100,\ (\ :5),\ 0.669 - 0.743i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.11520 + 0.496513i\)
\(L(\frac12)\) \(\approx\) \(1.11520 + 0.496513i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-13.0 - 29.2i)T \)
5 \( 1 \)
good3 \( 1 - 448. iT - 5.90e4T^{2} \)
7 \( 1 + 1.93e4iT - 2.82e8T^{2} \)
11 \( 1 + 2.07e5iT - 2.59e10T^{2} \)
13 \( 1 + 6.51e4T + 1.37e11T^{2} \)
17 \( 1 - 8.05e5T + 2.01e12T^{2} \)
19 \( 1 - 4.08e6iT - 6.13e12T^{2} \)
23 \( 1 - 4.40e6iT - 4.14e13T^{2} \)
29 \( 1 + 1.06e7T + 4.20e14T^{2} \)
31 \( 1 + 2.90e6iT - 8.19e14T^{2} \)
37 \( 1 - 1.45e7T + 4.80e15T^{2} \)
41 \( 1 + 6.38e7T + 1.34e16T^{2} \)
43 \( 1 + 2.40e8iT - 2.16e16T^{2} \)
47 \( 1 + 1.51e8iT - 5.25e16T^{2} \)
53 \( 1 - 3.71e8T + 1.74e17T^{2} \)
59 \( 1 + 4.69e8iT - 5.11e17T^{2} \)
61 \( 1 + 7.97e8T + 7.13e17T^{2} \)
67 \( 1 + 8.39e8iT - 1.82e18T^{2} \)
71 \( 1 + 4.87e8iT - 3.25e18T^{2} \)
73 \( 1 - 3.05e9T + 4.29e18T^{2} \)
79 \( 1 + 4.58e9iT - 9.46e18T^{2} \)
83 \( 1 + 2.95e8iT - 1.55e19T^{2} \)
89 \( 1 + 3.76e9T + 3.11e19T^{2} \)
97 \( 1 - 1.37e10T + 7.37e19T^{2} \)
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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

−11.88574612489220571844181276431, −10.67747960849447209796567821432, −9.863933752900496301614464231246, −8.742852402258542153878393945026, −7.71147493175636885697953254311, −5.99296793701387344096906139566, −5.13756008214463065023119135726, −3.79486635358132493212363338324, −3.50069664978275279730998722619, −0.30240381307754578558069599178, 1.01850425960262197806090115982, 2.15472289192944997883100728862, 2.75526475162428456515731465267, 4.91688929300284761040271638829, 6.06466969350288391218435578071, 7.17069387064627983075040578146, 8.499373778688684385165215241968, 9.515141844540773002598741349061, 11.19686020621701252311290225916, 12.07937236622505259170722703084

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