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Properties

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

\(L(\frac{1}{2})\) \(\approx\) \(0.8599980639\)
\(L(\frac12)\) \(\approx\) \(0.8599980639\)
\(L(1)\) 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 \( 1 + 0.618iT - T^{2} \)
17 \( 1 + 1.61iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - 1.61T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 1.61iT - T^{2} \)
41 \( 1 - 0.618T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 0.618iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 0.618T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 0.618iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - 1.61T + T^{2} \)
97 \( 1 + 1.61iT - T^{2} \)
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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

−8.966844480668144748648531848841, −8.249922078591038801952009196817, −7.59186397913319923042942574990, −6.81221823988510534100444383719, −6.01201859067437568955349665841, −5.28402419773630207685451165134, −4.65568841957767370800265542642, −3.46840180422705486251243593380, −2.60856274578672415848975273332, −0.62828097749147127048723355992, 1.32037444264783487904565903411, 2.43083820172957114867873953669, 3.27777613987017360413116688629, 4.18288672127242375762140413149, 4.97651031618491700260342894964, 5.94066651665368945124182990021, 6.61735618993872177850138455123, 8.065747707974702558488042021195, 8.377133069776784566232893356696, 9.154101812059798131991724460705

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