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Properties

Label 2-1925-385.69-c0-0-1
Degree $2$
Conductor $1925$
Sign $0.228 + 0.973i$
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  + (−0.786 − 1.08i)2-s + (−0.244 + 0.752i)4-s + (0.951 + 0.309i)7-s + (−0.266 + 0.0864i)8-s + (0.809 − 0.587i)9-s + (0.913 − 0.406i)11-s + (−0.413 − 1.27i)14-s + (0.942 + 0.684i)16-s + (−1.27 − 0.413i)18-s + (−1.15 − 0.669i)22-s + 1.95i·23-s + (−0.464 + 0.639i)28-s + (−0.564 + 1.73i)29-s − 1.27i·32-s + (0.244 + 0.752i)36-s + (1.27 + 0.413i)37-s + ⋯
L(s)  = 1  + (−0.786 − 1.08i)2-s + (−0.244 + 0.752i)4-s + (0.951 + 0.309i)7-s + (−0.266 + 0.0864i)8-s + (0.809 − 0.587i)9-s + (0.913 − 0.406i)11-s + (−0.413 − 1.27i)14-s + (0.942 + 0.684i)16-s + (−1.27 − 0.413i)18-s + (−1.15 − 0.669i)22-s + 1.95i·23-s + (−0.464 + 0.639i)28-s + (−0.564 + 1.73i)29-s − 1.27i·32-s + (0.244 + 0.752i)36-s + (1.27 + 0.413i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.228 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.228 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.228 + 0.973i$
Analytic conductor: \(0.960700\)
Root analytic conductor: \(0.980153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :0),\ 0.228 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9330646430\)
\(L(\frac12)\) \(\approx\) \(0.9330646430\)
\(L(1)\) 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 \( 1 + (-0.951 - 0.309i)T \)
11 \( 1 + (-0.913 + 0.406i)T \)
good2 \( 1 + (0.786 + 1.08i)T + (-0.309 + 0.951i)T^{2} \)
3 \( 1 + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 - 1.95iT - T^{2} \)
29 \( 1 + (0.564 - 1.73i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-1.27 - 0.413i)T + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 + 1.82iT - T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + 0.209iT - T^{2} \)
71 \( 1 + (1.47 + 1.07i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.169 - 0.122i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.309 + 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.309 - 0.951i)T^{2} \)
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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

−9.225652560943077756071077638400, −8.848016179393014651922822568319, −7.87904140875364461688527692137, −7.06105459265265385849685093454, −6.01295197131777820808283029471, −5.13799514709139894371353187221, −3.91781509833830119228549582157, −3.23713948822620039362399249890, −1.81078170087348822665596660449, −1.26139877507412360972888726633, 1.19679896467944076338305182802, 2.49441316282711766809340738063, 4.20398776046277766638072173817, 4.61529202109412784567603619420, 5.90890893062729154487223751241, 6.56180459068783493838888025385, 7.41909643249360629358834014639, 7.87466719378392552099934638346, 8.555153281573854353632950599326, 9.431259021209767936256284706855

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