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Properties

Label 2-1250-5.4-c1-0-39
Degree $2$
Conductor $1250$
Sign $-i$
Analytic cond. $9.98130$
Root an. cond. $3.15931$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − 2.33i·3-s − 4-s − 2.33·6-s − 3.77i·7-s + i·8-s − 2.44·9-s − 3.77·11-s + 2.33i·12-s + 3.17i·13-s − 3.77·14-s + 16-s + 1.39i·17-s + 2.44i·18-s − 3.91·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.34i·3-s − 0.5·4-s − 0.952·6-s − 1.42i·7-s + 0.353i·8-s − 0.813·9-s − 1.13·11-s + 0.673i·12-s + 0.880i·13-s − 1.00·14-s + 0.250·16-s + 0.337i·17-s + 0.575i·18-s − 0.897·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1250\)    =    \(2 \cdot 5^{4}\)
Sign: $-i$
Analytic conductor: \(9.98130\)
Root analytic conductor: \(3.15931\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1250} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1250,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6029163574\)
\(L(\frac12)\) \(\approx\) \(0.6029163574\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 \)
good3 \( 1 + 2.33iT - 3T^{2} \)
7 \( 1 + 3.77iT - 7T^{2} \)
11 \( 1 + 3.77T + 11T^{2} \)
13 \( 1 - 3.17iT - 13T^{2} \)
17 \( 1 - 1.39iT - 17T^{2} \)
19 \( 1 + 3.91T + 19T^{2} \)
23 \( 1 - 0.891iT - 23T^{2} \)
29 \( 1 + 0.0492T + 29T^{2} \)
31 \( 1 + 5.58T + 31T^{2} \)
37 \( 1 + 7.04iT - 37T^{2} \)
41 \( 1 + 1.48T + 41T^{2} \)
43 \( 1 - 2.69iT - 43T^{2} \)
47 \( 1 + 3.77iT - 47T^{2} \)
53 \( 1 - 11.6iT - 53T^{2} \)
59 \( 1 - 0.690T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 + 15.3iT - 67T^{2} \)
71 \( 1 + 6.69T + 71T^{2} \)
73 \( 1 - 5.16iT - 73T^{2} \)
79 \( 1 - 1.80T + 79T^{2} \)
83 \( 1 + 9.96iT - 83T^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 - 0.0901iT - 97T^{2} \)
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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

−9.067838866083934130626470187517, −8.101370364348976314769848180234, −7.44590800936176652091907659909, −6.84646412774530625928266265834, −5.82663704035455348451481913253, −4.57530261882743672337053917682, −3.72491935608904419135264301627, −2.38035760877541155612847155274, −1.51380370547656573594467768799, −0.24836444855857479739218596089, 2.45670601016537798690751687996, 3.42251820685545613392792334096, 4.60888576979316403864732707205, 5.34156877367537105621374772150, 5.75985425673973719402141428030, 6.98639266574753979892187461693, 8.207868806238633547190991324282, 8.589389006286831771003185738484, 9.485854317531713853303219117763, 10.12873315199944644523578384527

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