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Properties

Label 2-10e2-4.3-c10-0-75
Degree $2$
Conductor $100$
Sign $0.999 - 0.0189i$
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 − 0.302i)2-s + 375. i·3-s + (1.02e3 − 19.3i)4-s + (113. + 1.20e4i)6-s − 1.96e4i·7-s + (3.27e4 − 929. i)8-s − 8.21e4·9-s − 1.63e5i·11-s + (7.28e3 + 3.84e5i)12-s + 8.65e4·13-s + (−5.94e3 − 6.28e5i)14-s + (1.04e6 − 3.96e4i)16-s + 2.09e6·17-s + (−2.62e6 + 2.48e4i)18-s − 3.13e6i·19-s + ⋯
L(s)  = 1  + (0.999 − 0.00946i)2-s + 1.54i·3-s + (0.999 − 0.0189i)4-s + (0.0146 + 1.54i)6-s − 1.16i·7-s + (0.999 − 0.0283i)8-s − 1.39·9-s − 1.01i·11-s + (0.0292 + 1.54i)12-s + 0.233·13-s + (−0.0110 − 1.16i)14-s + (0.999 − 0.0378i)16-s + 1.47·17-s + (−1.39 + 0.0131i)18-s − 1.26i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0189i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.999 - 0.0189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.999 - 0.0189i$
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: \((2,\ 100,\ (\ :5),\ 0.999 - 0.0189i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(4.40319 + 0.0416564i\)
\(L(\frac12)\) \(\approx\) \(4.40319 + 0.0416564i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-31.9 + 0.302i)T \)
5 \( 1 \)
good3 \( 1 - 375. iT - 5.90e4T^{2} \)
7 \( 1 + 1.96e4iT - 2.82e8T^{2} \)
11 \( 1 + 1.63e5iT - 2.59e10T^{2} \)
13 \( 1 - 8.65e4T + 1.37e11T^{2} \)
17 \( 1 - 2.09e6T + 2.01e12T^{2} \)
19 \( 1 + 3.13e6iT - 6.13e12T^{2} \)
23 \( 1 + 1.05e7iT - 4.14e13T^{2} \)
29 \( 1 + 1.13e7T + 4.20e14T^{2} \)
31 \( 1 - 1.21e7iT - 8.19e14T^{2} \)
37 \( 1 + 4.87e7T + 4.80e15T^{2} \)
41 \( 1 - 4.55e7T + 1.34e16T^{2} \)
43 \( 1 + 7.47e7iT - 2.16e16T^{2} \)
47 \( 1 - 9.62e6iT - 5.25e16T^{2} \)
53 \( 1 - 1.27e8T + 1.74e17T^{2} \)
59 \( 1 - 1.18e9iT - 5.11e17T^{2} \)
61 \( 1 - 2.97e8T + 7.13e17T^{2} \)
67 \( 1 - 7.15e8iT - 1.82e18T^{2} \)
71 \( 1 - 2.56e9iT - 3.25e18T^{2} \)
73 \( 1 + 2.10e9T + 4.29e18T^{2} \)
79 \( 1 + 3.44e9iT - 9.46e18T^{2} \)
83 \( 1 + 7.09e9iT - 1.55e19T^{2} \)
89 \( 1 - 3.39e9T + 3.11e19T^{2} \)
97 \( 1 + 5.71e9T + 7.37e19T^{2} \)
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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

−11.68189953897895883267628170571, −10.66175224922906328859076520206, −10.27175885581075024532609369048, −8.726325553296924755654543529242, −7.22420811394149441309150058441, −5.80340466855990529046312748823, −4.72265273100266760265082050833, −3.83359636146126814289459979882, −2.98403945024454583498199485965, −0.78564163264230946042565387341, 1.41648773959947060796912372000, 2.11003452486969223720668849722, 3.44884794809544406166046752669, 5.38095547233717701464394333272, 6.07857947194725569666927693304, 7.33118077184928584669945802960, 8.024160193328300207799567667774, 9.780862019266421288779122292320, 11.49327907521638440861274324152, 12.28360235169747468105567550802

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