Properties

Label 2-45e2-9.5-c0-0-2
Degree $2$
Conductor $2025$
Sign $0.984 - 0.173i$
Analytic cond. $1.01060$
Root an. cond. $1.00528$
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  + (−0.866 + 0.5i)2-s i·8-s + (0.5 + 0.866i)16-s i·17-s + 19-s + (−0.866 − 0.5i)23-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)34-s + (−0.866 + 0.5i)38-s + 0.999·46-s + (1.73 − i)47-s + (0.5 − 0.866i)49-s + i·53-s + (0.5 + 0.866i)61-s + 0.999i·62-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s i·8-s + (0.5 + 0.866i)16-s i·17-s + 19-s + (−0.866 − 0.5i)23-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)34-s + (−0.866 + 0.5i)38-s + 0.999·46-s + (1.73 − i)47-s + (0.5 − 0.866i)49-s + i·53-s + (0.5 + 0.866i)61-s + 0.999i·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(1.01060\)
Root analytic conductor: \(1.00528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2025} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2025,\ (\ :0),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6695687030\)
\(L(\frac12)\) \(\approx\) \(0.6695687030\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 - iT - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−9.177953539975310770505828187088, −8.659129656626940218986409012421, −7.69269835053270633829700349082, −7.32760914496690127088645365632, −6.41247542612574389861234128499, −5.53790045439090630797049024542, −4.45715624614404614809135921284, −3.57076606916853206557710788218, −2.42929120577766331274259350851, −0.796436407443435296753350459483, 1.13816103219755869098658394721, 2.15575717556613149128907223841, 3.28404949262021698157272904817, 4.38599537733489761626788181243, 5.41546441004578799070022287827, 6.06028525691658715527193952385, 7.20526376663583571598193066095, 8.023021476433413874483974429290, 8.606411855580924525124675433534, 9.449518229671297645416639030100

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