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Properties

Label 2-1250-5.4-c1-0-8
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 + 2.61i·3-s − 4-s + 2.61·6-s − 3i·7-s + i·8-s − 3.85·9-s − 0.236·11-s − 2.61i·12-s − i·13-s − 3·14-s + 16-s + 7.85i·17-s + 3.85i·18-s + 0.854·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.51i·3-s − 0.5·4-s + 1.06·6-s − 1.13i·7-s + 0.353i·8-s − 1.28·9-s − 0.0711·11-s − 0.755i·12-s − 0.277i·13-s − 0.801·14-s + 0.250·16-s + 1.90i·17-s + 0.908i·18-s + 0.195·19-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

\(L(1)\) \(\approx\) \(1.190288697\)
\(L(\frac12)\) \(\approx\) \(1.190288697\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.61iT - 3T^{2} \)
7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 + 0.236T + 11T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 - 7.85iT - 17T^{2} \)
19 \( 1 - 0.854T + 19T^{2} \)
23 \( 1 - 6.23iT - 23T^{2} \)
29 \( 1 - 0.527T + 29T^{2} \)
31 \( 1 - 4.23T + 31T^{2} \)
37 \( 1 - 7.32iT - 37T^{2} \)
41 \( 1 + 7.47T + 41T^{2} \)
43 \( 1 - 1.76iT - 43T^{2} \)
47 \( 1 - 5.94iT - 47T^{2} \)
53 \( 1 + 1.52iT - 53T^{2} \)
59 \( 1 + 4.47T + 59T^{2} \)
61 \( 1 + 2.14T + 61T^{2} \)
67 \( 1 + 5.76iT - 67T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 - 5.70iT - 73T^{2} \)
79 \( 1 - 2.76T + 79T^{2} \)
83 \( 1 - 13.4iT - 83T^{2} \)
89 \( 1 - 4.47T + 89T^{2} \)
97 \( 1 + 10.5iT - 97T^{2} \)
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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

−10.09853308139901883055105431172, −9.459555537340097026532140779999, −8.472999647219883020959350964062, −7.75322934280391089100355959982, −6.41669150895263320496307641563, −5.34499805655237275091365696521, −4.47791658744744135279791929013, −3.80453512897240379575696678717, −3.16091563515137461281044982963, −1.42045734999397464307731722005, 0.52906112539768869495374409748, 2.09094740760426338672724572724, 2.94502519236222487858372176874, 4.67790530456344968264604590700, 5.54270491105238344165095561711, 6.35547004316027443783689250087, 7.02686640561711295193304630105, 7.66715392203154004831288775087, 8.600280185003600475092999139677, 9.055006705008655998124319974891

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