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Properties

Label 2-45e2-45.22-c0-0-1
Degree $2$
Conductor $2025$
Sign $0.216 - 0.976i$
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)4-s + (−0.448 + 1.67i)7-s + (0.499 + 0.866i)16-s − i·19-s + (−1.22 + 1.22i)28-s + (−0.5 + 0.866i)31-s + (1.22 + 1.22i)37-s + (−1.67 − 0.448i)43-s + (−1.73 − 1.00i)49-s + (0.5 + 0.866i)61-s + 0.999i·64-s + (1.22 − 1.22i)73-s + (0.5 − 0.866i)76-s + (0.866 − 0.5i)79-s + (0.448 − 1.67i)97-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)4-s + (−0.448 + 1.67i)7-s + (0.499 + 0.866i)16-s − i·19-s + (−1.22 + 1.22i)28-s + (−0.5 + 0.866i)31-s + (1.22 + 1.22i)37-s + (−1.67 − 0.448i)43-s + (−1.73 − 1.00i)49-s + (0.5 + 0.866i)61-s + 0.999i·64-s + (1.22 − 1.22i)73-s + (0.5 − 0.866i)76-s + (0.866 − 0.5i)79-s + (0.448 − 1.67i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $0.216 - 0.976i$
Analytic conductor: \(1.01060\)
Root analytic conductor: \(1.00528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2025} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2025,\ (\ :0),\ 0.216 - 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.314372601\)
\(L(\frac12)\) \(\approx\) \(1.314372601\)
\(L(1)\) 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 \( 1 + (-0.866 - 0.5i)T^{2} \)
7 \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + iT - T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
79 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.866 + 0.5i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2} \)
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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

−9.366881290545868701977732367091, −8.672702434857520957710557346683, −8.070129346688378317903550237092, −7.00644674748493869490837880357, −6.42402421049463926372841463199, −5.65323151375507495814706476380, −4.77099132447632565014341552852, −3.35782550726286062781193713082, −2.74933655096878687748595875236, −1.86276486528210039305198006380, 0.946009940028490587731222107555, 2.12711881222540159272032594519, 3.40710895567187860989657781382, 4.09268397827519648178573882438, 5.24228943201412607841792366143, 6.20030081800276178073268841462, 6.78446880309423343464631546032, 7.54918943151303961872585412896, 8.086442419154117899633623208479, 9.557081103902497102351016935684

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