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Properties

Label 2-45e2-45.7-c0-0-2
Degree $2$
Conductor $2025$
Sign $0.999 + 0.0438i$
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 + (1.67 − 0.448i)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 + (0.448 + 1.67i)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 + (−1.67 + 0.448i)97-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)4-s + (1.67 − 0.448i)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 + (0.448 + 1.67i)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 + (−1.67 + 0.448i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $0.999 + 0.0438i$
Analytic conductor: \(1.01060\)
Root analytic conductor: \(1.00528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2025} (1432, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2025,\ (\ :0),\ 0.999 + 0.0438i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.152325098\)
\(L(\frac12)\) \(\approx\) \(1.152325098\)
\(L(1)\) 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 \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \)
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 \( 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2} \)
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 \( 1 + (1.67 - 0.448i)T + (0.866 - 0.5i)T^{2} \)
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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

−9.250268966268234403334913311055, −8.414017307325871895772966280758, −7.88616168199569739273788478554, −7.30207478789387664035889052363, −6.09390801263172816929505254309, −4.88015897349262687325246178872, −4.68316913927023268363890228736, −3.70377690339327215326203808305, −2.47583106525465690559135112607, −1.09699256581154718568168489742, 1.26734063687963245963115861840, 2.23017344388341383376505677678, 3.81726409829568152142936679061, 4.51692093088414541889381651128, 5.43212924191603903644045174762, 5.75622041814312355826131836706, 7.13146079529074366842506003617, 8.018211899221571245435642590047, 8.552115691129193745667829730372, 9.199704261167514198116804658283

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