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Properties

Label 2-45e2-225.86-c0-0-1
Degree $2$
Conductor $2025$
Sign $-0.872 + 0.487i$
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  + (0.658 − 1.47i)2-s + (−1.08 − 1.20i)4-s + (−0.406 − 0.913i)5-s + (0.809 + 1.40i)7-s + (−0.951 + 0.309i)8-s − 1.61·10-s + (0.406 − 0.913i)11-s + (2.60 − 0.273i)14-s + (0.951 − 0.309i)17-s + (−0.309 − 0.951i)19-s + (−0.658 + 1.47i)20-s + (−1.08 − 1.20i)22-s + (−0.994 + 0.104i)23-s + (−0.669 + 0.743i)25-s + (0.809 − 2.48i)28-s + (−0.128 − 0.604i)29-s + ⋯
L(s)  = 1  + (0.658 − 1.47i)2-s + (−1.08 − 1.20i)4-s + (−0.406 − 0.913i)5-s + (0.809 + 1.40i)7-s + (−0.951 + 0.309i)8-s − 1.61·10-s + (0.406 − 0.913i)11-s + (2.60 − 0.273i)14-s + (0.951 − 0.309i)17-s + (−0.309 − 0.951i)19-s + (−0.658 + 1.47i)20-s + (−1.08 − 1.20i)22-s + (−0.994 + 0.104i)23-s + (−0.669 + 0.743i)25-s + (0.809 − 2.48i)28-s + (−0.128 − 0.604i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $-0.872 + 0.487i$
Analytic conductor: \(1.01060\)
Root analytic conductor: \(1.00528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2025} (1511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2025,\ (\ :0),\ -0.872 + 0.487i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.617378070\)
\(L(\frac12)\) \(\approx\) \(1.617378070\)
\(L(1)\) 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 \( 1 + (0.406 + 0.913i)T \)
good2 \( 1 + (-0.658 + 1.47i)T + (-0.669 - 0.743i)T^{2} \)
7 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.406 + 0.913i)T + (-0.669 - 0.743i)T^{2} \)
13 \( 1 + (0.669 - 0.743i)T^{2} \)
17 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.994 - 0.104i)T + (0.978 - 0.207i)T^{2} \)
29 \( 1 + (0.128 + 0.604i)T + (-0.913 + 0.406i)T^{2} \)
31 \( 1 + (0.913 + 0.406i)T^{2} \)
37 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.406 - 0.913i)T + (-0.669 + 0.743i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.128 - 0.604i)T + (-0.913 + 0.406i)T^{2} \)
53 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.669 + 0.743i)T^{2} \)
61 \( 1 + (0.913 + 0.406i)T + (0.669 + 0.743i)T^{2} \)
67 \( 1 + (-1.58 - 0.336i)T + (0.913 + 0.406i)T^{2} \)
71 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.604 + 0.128i)T + (0.913 - 0.406i)T^{2} \)
83 \( 1 + (-1.20 - 1.08i)T + (0.104 + 0.994i)T^{2} \)
89 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.978 - 0.207i)T + (0.913 - 0.406i)T^{2} \)
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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

−9.366324793069934255621070039832, −8.311627170893423319566816436771, −7.905877864815911635096239493840, −6.21967472153827204496766055983, −5.39284091487447534990618598929, −4.84499791008365961274383840133, −3.98921884107341092260089574970, −3.02181317791821850648946541618, −2.10726935284601061721497614794, −1.05855294438552022672039004866, 1.77185956961118158341744492544, 3.61535207302980144609769863951, 4.05097857367304250921367490676, 4.82411197110975556831940292964, 5.88163008675857921057849790929, 6.60881408818898429370450633879, 7.36378058099552671984512556327, 7.71724512133288182218206897259, 8.290964416426747281968670875001, 9.648855662409222627957577674623

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