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Properties

Label 2-1925-385.104-c0-0-4
Degree $2$
Conductor $1925$
Sign $0.599 + 0.800i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.599 + 0.800i$
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: \((2,\ 1925,\ (\ :0),\ 0.599 + 0.800i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.081274422\)
\(L(\frac12)\) \(\approx\) \(3.081274422\)
\(L(1)\) 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

−9.408402175461635104168845408621, −8.515213268600562484106743402199, −7.45235242619962003870173683508, −6.54158563694612814384119405863, −5.69906048202136760575824640641, −5.23943973779034574364222728582, −4.37859341780840517411889648550, −3.38169392908944088262061741264, −2.65550097426168998035008162250, −1.67680254133498488568012678783, 2.04014203611695339093559670541, 2.93326725524843333581065944847, 4.02392931139826292545670122743, 4.64545440392977050167704905473, 5.45091122817873632303688626323, 5.98051167550570332304037136841, 7.11886024847738284655650533169, 7.80841366505452868529395436789, 8.050787770801778873349041362928, 9.565573908369971118615033437682

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