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Properties

Label 2-1925-385.272-c0-0-7
Degree $2$
Conductor $1925$
Sign $-0.0128 + 0.999i$
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  + (0.863 − 1.69i)3-s + (0.951 + 0.309i)4-s + (−0.891 + 0.453i)7-s + (−1.53 − 2.11i)9-s + (0.809 − 0.587i)11-s + (1.34 − 1.34i)12-s + (−0.0966 + 0.610i)13-s + (0.809 + 0.587i)16-s + (−0.253 − 1.59i)17-s + 1.90i·21-s + (−3.03 + 0.481i)27-s + (−0.987 + 0.156i)28-s + (−0.363 + 1.11i)29-s + (−0.297 − 1.87i)33-s + (−0.809 − 2.48i)36-s + ⋯
L(s)  = 1  + (0.863 − 1.69i)3-s + (0.951 + 0.309i)4-s + (−0.891 + 0.453i)7-s + (−1.53 − 2.11i)9-s + (0.809 − 0.587i)11-s + (1.34 − 1.34i)12-s + (−0.0966 + 0.610i)13-s + (0.809 + 0.587i)16-s + (−0.253 − 1.59i)17-s + 1.90i·21-s + (−3.03 + 0.481i)27-s + (−0.987 + 0.156i)28-s + (−0.363 + 1.11i)29-s + (−0.297 − 1.87i)33-s + (−0.809 − 2.48i)36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $-0.0128 + 0.999i$
Analytic conductor: \(0.960700\)
Root analytic conductor: \(0.980153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :0),\ -0.0128 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.719771661\)
\(L(\frac12)\) \(\approx\) \(1.719771661\)
\(L(1)\) 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 \( 1 + (0.891 - 0.453i)T \)
11 \( 1 + (-0.809 + 0.587i)T \)
good2 \( 1 + (-0.951 - 0.309i)T^{2} \)
3 \( 1 + (-0.863 + 1.69i)T + (-0.587 - 0.809i)T^{2} \)
13 \( 1 + (0.0966 - 0.610i)T + (-0.951 - 0.309i)T^{2} \)
17 \( 1 + (0.253 + 1.59i)T + (-0.951 + 0.309i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.587 - 0.809i)T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-1.04 - 0.533i)T + (0.587 + 0.809i)T^{2} \)
53 \( 1 + (-0.951 - 0.309i)T^{2} \)
59 \( 1 + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.309 + 0.951i)T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.734 - 1.44i)T + (-0.587 + 0.809i)T^{2} \)
79 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.610 + 0.0966i)T + (0.951 - 0.309i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.183 - 1.16i)T + (-0.951 - 0.309i)T^{2} \)
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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

−9.025228208868335447516625583222, −8.363448074192869900064087399351, −7.35246326555916851622555028996, −6.94536936455494400995844571750, −6.42576311410741793736479513687, −5.61832203421280956164168610217, −3.72704659912055508208102821827, −2.91893958113262357524410322942, −2.34727486058073110445552667004, −1.20715323174183806367241544884, 1.98342010386541193676685383632, 2.99532687784710970951164703561, 3.77731618361856910372072010553, 4.37877260366601211433009322469, 5.61204357121730732635222048266, 6.28194290463954634161192099309, 7.33848704080996248530053691954, 8.146842588036899193075211335404, 9.034907229914865757382412754551, 9.724563028064299728879807534534

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