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Properties

Label 2-50e2-20.19-c0-0-0
Degree $2$
Conductor $2500$
Sign $-1$
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  + i·2-s − 4-s − i·8-s − 9-s + 1.61i·13-s + 16-s + 0.618i·17-s − i·18-s − 1.61·26-s − 0.618·29-s + i·32-s − 0.618·34-s + 36-s + 0.618i·37-s − 1.61·41-s + ⋯
L(s)  = 1  + i·2-s − 4-s − i·8-s − 9-s + 1.61i·13-s + 16-s + 0.618i·17-s − i·18-s − 1.61·26-s − 0.618·29-s + i·32-s − 0.618·34-s + 36-s + 0.618i·37-s − 1.61·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $-1$
Analytic conductor: \(1.24766\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2500} (2499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2500,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6273521998\)
\(L(\frac12)\) \(\approx\) \(0.6273521998\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.201211723692178182744669660646, −8.665275005434472503828991914548, −7.997720654714275675659656026914, −7.08815877152852286838999792496, −6.41578702818397139456434879587, −5.77882255759706182447512753027, −4.86412263691690495300321104041, −4.10149693836474116741102207030, −3.13289193927527128182397557589, −1.67676929632192092176702907719, 0.40955603311560012748092629782, 1.94002225091826912362541592745, 3.04799198685759782132689532728, 3.44772444460552869119794460845, 4.80072268899660433167958533506, 5.38646890034592018374366773567, 6.14759707981169934111053193586, 7.46291817633939127720709648845, 8.185448202910269074359029374344, 8.776350118443228123755010972835

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