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Properties

Label 2-50e2-2500.1191-c0-0-0
Degree $2$
Conductor $2500$
Sign $-0.00753 - 0.999i$
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  + (−0.745 + 0.666i)2-s + (0.112 − 0.993i)4-s + (−0.379 + 0.925i)5-s + (0.577 + 0.816i)8-s + (0.448 + 0.893i)9-s + (−0.332 − 0.942i)10-s + (0.669 − 0.661i)13-s + (−0.974 − 0.224i)16-s + (1.29 − 0.589i)17-s + (−0.929 − 0.368i)18-s + (0.876 + 0.481i)20-s + (−0.711 − 0.702i)25-s + (−0.0591 + 0.939i)26-s + (0.480 + 1.77i)29-s + (0.876 − 0.481i)32-s + ⋯
L(s)  = 1  + (−0.745 + 0.666i)2-s + (0.112 − 0.993i)4-s + (−0.379 + 0.925i)5-s + (0.577 + 0.816i)8-s + (0.448 + 0.893i)9-s + (−0.332 − 0.942i)10-s + (0.669 − 0.661i)13-s + (−0.974 − 0.224i)16-s + (1.29 − 0.589i)17-s + (−0.929 − 0.368i)18-s + (0.876 + 0.481i)20-s + (−0.711 − 0.702i)25-s + (−0.0591 + 0.939i)26-s + (0.480 + 1.77i)29-s + (0.876 − 0.481i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00753 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00753 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $-0.00753 - 0.999i$
Analytic conductor: \(1.24766\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2500} (1191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2500,\ (\ :0),\ -0.00753 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8436126133\)
\(L(\frac12)\) \(\approx\) \(0.8436126133\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.745 - 0.666i)T \)
5 \( 1 + (0.379 - 0.925i)T \)
good3 \( 1 + (-0.448 - 0.893i)T^{2} \)
7 \( 1 + (0.992 - 0.125i)T^{2} \)
11 \( 1 + (0.910 + 0.414i)T^{2} \)
13 \( 1 + (-0.669 + 0.661i)T + (0.0125 - 0.999i)T^{2} \)
17 \( 1 + (-1.29 + 0.589i)T + (0.656 - 0.754i)T^{2} \)
19 \( 1 + (-0.988 - 0.150i)T^{2} \)
23 \( 1 + (-0.998 - 0.0502i)T^{2} \)
29 \( 1 + (-0.480 - 1.77i)T + (-0.863 + 0.503i)T^{2} \)
31 \( 1 + (-0.656 + 0.754i)T^{2} \)
37 \( 1 + (-0.342 + 0.570i)T + (-0.470 - 0.882i)T^{2} \)
41 \( 1 + (-0.782 + 0.599i)T + (0.260 - 0.965i)T^{2} \)
43 \( 1 + (-0.0627 + 0.998i)T^{2} \)
47 \( 1 + (0.999 - 0.0251i)T^{2} \)
53 \( 1 + (1.50 - 0.732i)T + (0.617 - 0.786i)T^{2} \)
59 \( 1 + (-0.850 + 0.525i)T^{2} \)
61 \( 1 + (-1.21 - 0.927i)T + (0.260 + 0.965i)T^{2} \)
67 \( 1 + (-0.994 + 0.100i)T^{2} \)
71 \( 1 + (-0.793 + 0.607i)T^{2} \)
73 \( 1 + (0.0435 + 1.15i)T + (-0.997 + 0.0753i)T^{2} \)
79 \( 1 + (-0.448 - 0.893i)T^{2} \)
83 \( 1 + (0.711 + 0.702i)T^{2} \)
89 \( 1 + (-1.22 - 1.56i)T + (-0.236 + 0.971i)T^{2} \)
97 \( 1 + (1.03 - 0.675i)T + (0.402 - 0.915i)T^{2} \)
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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

−9.272079835531540908123511621005, −8.290977984175828980764167069780, −7.70463042289082235608393209074, −7.23572830738364316954042179107, −6.38809067406208770836990656854, −5.55551230510830577344100670550, −4.80872734467324639696690099077, −3.55171205964292466957378038835, −2.56486793585113752118404080017, −1.25062384037082303602000968582, 0.883466813355813699221398958992, 1.74926209533075267843142607351, 3.22164666157043875093901135533, 3.95734399744709779004576380481, 4.61781397204021332046616024430, 5.95572107372209422995486674796, 6.69938683764810463850016948001, 7.80628362133407305909471068263, 8.194415873621778615439820401087, 8.981380846828239915197492088798

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