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Properties

Label 2-50e2-100.19-c0-0-5
Degree $2$
Conductor $2500$
Sign $0.929 - 0.368i$
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  + (0.309 + 0.951i)2-s + (1.30 − 0.951i)3-s + (−0.809 + 0.587i)4-s + (1.30 + 0.951i)6-s + 0.618·7-s + (−0.809 − 0.587i)8-s + (0.500 − 1.53i)9-s + (−0.499 + 1.53i)12-s + (0.190 + 0.587i)14-s + (0.309 − 0.951i)16-s + 1.61·18-s + (0.809 − 0.587i)21-s + (0.190 + 0.587i)23-s − 1.61·24-s + (−0.309 − 0.951i)27-s + (−0.5 + 0.363i)28-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (1.30 − 0.951i)3-s + (−0.809 + 0.587i)4-s + (1.30 + 0.951i)6-s + 0.618·7-s + (−0.809 − 0.587i)8-s + (0.500 − 1.53i)9-s + (−0.499 + 1.53i)12-s + (0.190 + 0.587i)14-s + (0.309 − 0.951i)16-s + 1.61·18-s + (0.809 − 0.587i)21-s + (0.190 + 0.587i)23-s − 1.61·24-s + (−0.309 − 0.951i)27-s + (−0.5 + 0.363i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $0.929 - 0.368i$
Analytic conductor: \(1.24766\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2500} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2500,\ (\ :0),\ 0.929 - 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.106857549\)
\(L(\frac12)\) \(\approx\) \(2.106857549\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 \)
good3 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 - 0.618T + T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
29 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + 1.61T + T^{2} \)
47 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T^{2} \)
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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

−8.810391559927406392280941667808, −8.051686123880183562746876826449, −7.86699011838669710456389966770, −6.91221735529216945540713469867, −6.38795846868676664237777308976, −5.28137136850796864545374456611, −4.40351728567085114052135257959, −3.41831115214839061594637993902, −2.60656313457059585230521389098, −1.37174479817972640737164382247, 1.55779364290027309544582065148, 2.60471948355972654842936781932, 3.24323451761269354755063409380, 4.15743455975078875155715483799, 4.72131924279114865200428491182, 5.50609553873836388668406443944, 6.80957694720395977291560723387, 7.991739548550134701329758622322, 8.616178787537971930307833150584, 9.028238557505830479410873611361

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