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Properties

Label 2-1925-385.104-c0-0-0
Degree $2$
Conductor $1925$
Sign $-0.999 - 0.000991i$
Analytic cond. $0.960700$
Root an. cond. $0.980153$
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  + (−1.73 + 0.564i)2-s + (1.89 − 1.37i)4-s + (−0.587 − 0.809i)7-s + (−1.43 + 1.97i)8-s + (−0.309 − 0.951i)9-s + (−0.978 + 0.207i)11-s + (1.47 + 1.07i)14-s + (0.657 − 2.02i)16-s + (1.07 + 1.47i)18-s + (1.58 − 0.913i)22-s + 0.209i·23-s + (−2.22 − 0.722i)28-s + (−1.58 + 1.14i)29-s + 1.44i·32-s + (−1.89 − 1.37i)36-s + (−1.07 − 1.47i)37-s + ⋯
L(s)  = 1  + (−1.73 + 0.564i)2-s + (1.89 − 1.37i)4-s + (−0.587 − 0.809i)7-s + (−1.43 + 1.97i)8-s + (−0.309 − 0.951i)9-s + (−0.978 + 0.207i)11-s + (1.47 + 1.07i)14-s + (0.657 − 2.02i)16-s + (1.07 + 1.47i)18-s + (1.58 − 0.913i)22-s + 0.209i·23-s + (−2.22 − 0.722i)28-s + (−1.58 + 1.14i)29-s + 1.44i·32-s + (−1.89 − 1.37i)36-s + (−1.07 − 1.47i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.000991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.000991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $-0.999 - 0.000991i$
Analytic conductor: \(0.960700\)
Root analytic conductor: \(0.980153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (874, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :0),\ -0.999 - 0.000991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001351640349\)
\(L(\frac12)\) \(\approx\) \(0.001351640349\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + (0.587 + 0.809i)T \)
11 \( 1 + (0.978 - 0.207i)T \)
good2 \( 1 + (1.73 - 0.564i)T + (0.809 - 0.587i)T^{2} \)
3 \( 1 + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 - 0.209iT - T^{2} \)
29 \( 1 + (1.58 - 1.14i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (1.07 + 1.47i)T + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - 1.95iT - T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 - 1.33iT - T^{2} \)
71 \( 1 + (0.604 - 1.86i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.413 + 1.27i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.809 - 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.809 + 0.587i)T^{2} \)
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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

−9.588746998560558051684324422232, −9.074250288431770004331511252978, −8.281470413353889535865386136412, −7.35140670336762901500965128067, −7.07883744171203808208207251813, −6.12136770852779955096746276206, −5.39334685718314717474884900359, −3.84842242039509075559442924340, −2.74604465120735143124495747679, −1.37520616848157405049021944248, 0.00170324974573137404757170356, 1.93383126400212193222145153414, 2.55383660161100485887394525740, 3.47214918874396221410336250126, 5.12448204413061883462017144425, 5.96906412062836974639565172667, 7.01965619563069384234302314663, 7.78581274006977535039298450448, 8.375944203265355634863405581806, 8.989089179679707786925724009428

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