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Properties

Label 2-45e2-45.29-c0-0-1
Degree $2$
Conductor $2025$
Sign $0.917 - 0.397i$
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  + (0.5 + 0.866i)4-s + (−0.866 − 0.5i)7-s + (1.73 − i)13-s + (−0.499 + 0.866i)16-s + 19-s − 0.999i·28-s + (0.5 + 0.866i)31-s + i·37-s + (0.866 + 0.5i)43-s + (1.73 + 0.999i)52-s + (0.5 − 0.866i)61-s − 0.999·64-s + (−1.73 + i)67-s − i·73-s + (0.5 + 0.866i)76-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)4-s + (−0.866 − 0.5i)7-s + (1.73 − i)13-s + (−0.499 + 0.866i)16-s + 19-s − 0.999i·28-s + (0.5 + 0.866i)31-s + i·37-s + (0.866 + 0.5i)43-s + (1.73 + 0.999i)52-s + (0.5 − 0.866i)61-s − 0.999·64-s + (−1.73 + i)67-s − i·73-s + (0.5 + 0.866i)76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $0.917 - 0.397i$
Analytic conductor: \(1.01060\)
Root analytic conductor: \(1.00528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2025} (674, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2025,\ (\ :0),\ 0.917 - 0.397i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.298321952\)
\(L(\frac12)\) \(\approx\) \(1.298321952\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - iT - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + iT - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
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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

−9.318811398668117819633076283477, −8.439125676540347282566296845211, −7.898740605560842738238515172294, −7.00045604406334869669917411262, −6.37769374262913389587980367609, −5.57077216309624019353381112317, −4.24930505098907511555571760822, −3.34907027759168016892439827197, −2.96159028218511197966092612328, −1.27111970955611383956660261040, 1.18813921089393403007175366882, 2.33384816977556655199343047346, 3.38699022628688049771200366029, 4.37237449095198621933523325672, 5.64483421598684551117399624383, 6.04055870999586731794032196622, 6.74967848808302019581729487642, 7.58496833675444800750523992766, 8.798891862954446786313430683094, 9.263363718829593731497623773546

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