Properties

Label 2-1250-5.4-c1-0-4
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 − 2.96i·3-s − 4-s + 2.96·6-s + 1.83i·7-s i·8-s − 5.79·9-s − 1.83·11-s + 2.96i·12-s + 2.41i·13-s − 1.83·14-s + 16-s + 2.78i·17-s − 5.79i·18-s + 1.67·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.71i·3-s − 0.5·4-s + 1.21·6-s + 0.692i·7-s − 0.353i·8-s − 1.93·9-s − 0.552·11-s + 0.856i·12-s + 0.670i·13-s − 0.489·14-s + 0.250·16-s + 0.675i·17-s − 1.36i·18-s + 0.383·19-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

\(L(1)\) \(\approx\) \(0.8388660713\)
\(L(\frac12)\) \(\approx\) \(0.8388660713\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.96iT - 3T^{2} \)
7 \( 1 - 1.83iT - 7T^{2} \)
11 \( 1 + 1.83T + 11T^{2} \)
13 \( 1 - 2.41iT - 13T^{2} \)
17 \( 1 - 2.78iT - 17T^{2} \)
19 \( 1 - 1.67T + 19T^{2} \)
23 \( 1 - 7.76iT - 23T^{2} \)
29 \( 1 + 7.58T + 29T^{2} \)
31 \( 1 + 5.29T + 31T^{2} \)
37 \( 1 + 1.31iT - 37T^{2} \)
41 \( 1 - 3.51T + 41T^{2} \)
43 \( 1 - 4.30iT - 43T^{2} \)
47 \( 1 - 1.83iT - 47T^{2} \)
53 \( 1 - 6.51iT - 53T^{2} \)
59 \( 1 - 9.06T + 59T^{2} \)
61 \( 1 - 2.58T + 61T^{2} \)
67 \( 1 - 9.50iT - 67T^{2} \)
71 \( 1 - 0.305T + 71T^{2} \)
73 \( 1 + 14.9iT - 73T^{2} \)
79 \( 1 - 3.46T + 79T^{2} \)
83 \( 1 - 6.37iT - 83T^{2} \)
89 \( 1 - 3.32T + 89T^{2} \)
97 \( 1 - 11.0iT - 97T^{2} \)
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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

−9.466963597464925185934220393701, −8.908044444320560859532602920847, −7.88707030709441928906547585371, −7.52219638466885184364000152539, −6.71038170258528781460770362355, −5.80676652747203406981083604678, −5.38563825396175205026475998952, −3.75727419712785892919034017607, −2.41502187586504972719759957706, −1.42686658328713912079255982451, 0.35839607037691932125531672962, 2.48715653810262041909912497459, 3.46920833661267858839790320077, 4.14559232662747553593798641940, 5.04193299134563080051864341503, 5.63263049601691481806075423309, 7.14109496137102645565098015401, 8.198254606157045534592550748266, 8.994632709290590624955956805440, 9.764065146135516144399318650758

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