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Properties

Label 2-10e2-20.19-c8-0-58
Degree $2$
Conductor $100$
Sign $0.970 - 0.239i$
Analytic cond. $40.7378$
Root an. cond. $6.38262$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.70 + 14.5i)2-s + 150.·3-s + (−166. − 194. i)4-s + (−1.00e3 + 2.18e3i)6-s + 2.62e3·7-s + (3.94e3 − 1.10e3i)8-s + 1.60e4·9-s − 2.30e3i·11-s + (−2.49e4 − 2.92e4i)12-s − 4.70e4i·13-s + (−1.76e4 + 3.81e4i)14-s + (−1.03e4 + 6.47e4i)16-s − 5.19e4i·17-s + (−1.07e5 + 2.32e5i)18-s − 5.95e4i·19-s + ⋯
L(s)  = 1  + (−0.419 + 0.907i)2-s + 1.85·3-s + (−0.648 − 0.760i)4-s + (−0.777 + 1.68i)6-s + 1.09·7-s + (0.962 − 0.270i)8-s + 2.43·9-s − 0.157i·11-s + (−1.20 − 1.41i)12-s − 1.64i·13-s + (−0.458 + 0.993i)14-s + (−0.158 + 0.987i)16-s − 0.622i·17-s + (−1.02 + 2.21i)18-s − 0.456i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.970 - 0.239i$
Analytic conductor: \(40.7378\)
Root analytic conductor: \(6.38262\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :4),\ 0.970 - 0.239i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(3.54366 + 0.431496i\)
\(L(\frac12)\) \(\approx\) \(3.54366 + 0.431496i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (6.70 - 14.5i)T \)
5 \( 1 \)
good3 \( 1 - 150.T + 6.56e3T^{2} \)
7 \( 1 - 2.62e3T + 5.76e6T^{2} \)
11 \( 1 + 2.30e3iT - 2.14e8T^{2} \)
13 \( 1 + 4.70e4iT - 8.15e8T^{2} \)
17 \( 1 + 5.19e4iT - 6.97e9T^{2} \)
19 \( 1 + 5.95e4iT - 1.69e10T^{2} \)
23 \( 1 + 7.75e4T + 7.83e10T^{2} \)
29 \( 1 + 9.02e5T + 5.00e11T^{2} \)
31 \( 1 + 3.40e5iT - 8.52e11T^{2} \)
37 \( 1 + 5.84e5iT - 3.51e12T^{2} \)
41 \( 1 - 2.93e5T + 7.98e12T^{2} \)
43 \( 1 - 2.95e6T + 1.16e13T^{2} \)
47 \( 1 - 5.03e6T + 2.38e13T^{2} \)
53 \( 1 - 7.54e6iT - 6.22e13T^{2} \)
59 \( 1 - 8.82e6iT - 1.46e14T^{2} \)
61 \( 1 - 1.08e7T + 1.91e14T^{2} \)
67 \( 1 - 1.44e7T + 4.06e14T^{2} \)
71 \( 1 - 3.71e6iT - 6.45e14T^{2} \)
73 \( 1 - 3.62e7iT - 8.06e14T^{2} \)
79 \( 1 - 4.88e7iT - 1.51e15T^{2} \)
83 \( 1 + 6.93e7T + 2.25e15T^{2} \)
89 \( 1 - 1.05e8T + 3.93e15T^{2} \)
97 \( 1 - 1.33e8iT - 7.83e15T^{2} \)
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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

−12.87681391092379373901755268853, −10.82159268687218667505019647772, −9.680368802279580041277198755018, −8.740492487501033536950015850937, −7.919813241092990612288477242020, −7.34596511057705221542298831134, −5.39293994752630964640363961882, −4.02784504484660043159924958796, −2.46132149393240794053981338741, −1.01276972890145104885699823166, 1.60029062558274061845964462025, 2.11919391112315221314935783230, 3.65447007866916386990057409235, 4.48806319577560437993471787665, 7.28280728226040282860894137458, 8.195606446134402912722315830461, 8.959638471314190548910642290422, 9.799558647918403697403022048282, 11.06315550905771347121694704790, 12.25647109862868395437283314598

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