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Properties

Label 2-50e2-100.91-c0-0-5
Degree $2$
Conductor $2500$
Sign $-0.876 + 0.481i$
Analytic cond. $1.24766$
Root an. cond. $1.11698$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.587 − 0.809i)2-s + (0.587 − 0.190i)3-s + (−0.309 − 0.951i)4-s + (0.190 − 0.587i)6-s − 1.61i·7-s + (−0.951 − 0.309i)8-s + (−0.5 + 0.363i)9-s + (−0.363 − 0.5i)12-s + (−1.30 − 0.951i)14-s + (−0.809 + 0.587i)16-s + 0.618i·18-s + (−0.309 − 0.951i)21-s + (0.951 − 1.30i)23-s − 0.618·24-s + (−0.587 + 0.809i)27-s + (−1.53 + 0.500i)28-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)2-s + (0.587 − 0.190i)3-s + (−0.309 − 0.951i)4-s + (0.190 − 0.587i)6-s − 1.61i·7-s + (−0.951 − 0.309i)8-s + (−0.5 + 0.363i)9-s + (−0.363 − 0.5i)12-s + (−1.30 − 0.951i)14-s + (−0.809 + 0.587i)16-s + 0.618i·18-s + (−0.309 − 0.951i)21-s + (0.951 − 1.30i)23-s − 0.618·24-s + (−0.587 + 0.809i)27-s + (−1.53 + 0.500i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $-0.876 + 0.481i$
Analytic conductor: \(1.24766\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2500} (2251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2500,\ (\ :0),\ -0.876 + 0.481i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.665937012\)
\(L(\frac12)\) \(\approx\) \(1.665937012\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 + 0.809i)T \)
5 \( 1 \)
good3 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
7 \( 1 + 1.61iT - T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
29 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 + 0.618iT - T^{2} \)
47 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (-1.90 - 0.618i)T + (0.809 + 0.587i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.809 + 0.587i)T^{2} \)
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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

−8.794478669841762902550729506772, −8.141529285434252926058394791132, −7.17084799722529892139884710536, −6.54310678132291643503788433857, −5.41503716624526027082778941946, −4.58867946600312724042359510031, −3.84054596958444014590883947637, −3.03359652452544854717038050729, −2.10422275818471476070073347601, −0.844240041775444926857334628517, 2.17229308819596054034662148630, 3.08708582382879308065376025631, 3.65284084308167011222067309429, 4.99924633492714641685643618779, 5.47072798143852904862901344327, 6.24243236632931203653787735159, 7.03437866517609003305961189580, 8.089563309002026135739802278099, 8.514505139075205137528522965321, 9.278549809223708385866770131563

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