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Properties

Label 2-45e2-45.22-c0-0-4
Degree $2$
Conductor $2025$
Sign $-0.929 - 0.370i$
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  + (0.448 − 1.67i)2-s + (−1.73 − 1.00i)4-s + (−1.22 + 1.22i)8-s + (0.500 + 0.866i)16-s + (−1.22 − 1.22i)17-s − i·19-s + (−0.448 − 1.67i)23-s + (−0.5 + 0.866i)31-s + (−2.59 + 1.5i)34-s + (−1.67 − 0.448i)38-s − 3·46-s + (0.866 + 0.5i)49-s + (1.22 − 1.22i)53-s + (0.5 + 0.866i)61-s + (1.22 + 1.22i)62-s + ⋯
L(s)  = 1  + (0.448 − 1.67i)2-s + (−1.73 − 1.00i)4-s + (−1.22 + 1.22i)8-s + (0.500 + 0.866i)16-s + (−1.22 − 1.22i)17-s − i·19-s + (−0.448 − 1.67i)23-s + (−0.5 + 0.866i)31-s + (−2.59 + 1.5i)34-s + (−1.67 − 0.448i)38-s − 3·46-s + (0.866 + 0.5i)49-s + (1.22 − 1.22i)53-s + (0.5 + 0.866i)61-s + (1.22 + 1.22i)62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $-0.929 - 0.370i$
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: \((2,\ 2025,\ (\ :0),\ -0.929 - 0.370i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.085478904\)
\(L(\frac12)\) \(\approx\) \(1.085478904\)
\(L(1)\) 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 \( 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2} \)
7 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
19 \( 1 + iT - T^{2} \)
23 \( 1 + (0.448 + 1.67i)T + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
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 \( 1 - iT^{2} \)
79 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.866 - 0.5i)T^{2} \)
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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

−9.042421366577650095154028792539, −8.637123359811131116660890218610, −7.25335801888225744033730210682, −6.48371831000082354931490752321, −5.18855726304513610345139016795, −4.64462652694273341406441197563, −3.82420398345039536440586047557, −2.71203700727663074504828196325, −2.19840905532679268690175473520, −0.66474401758728934258476016921, 1.95832967104401859576301929090, 3.69919903941742720566032179578, 4.15315558916337440724122730744, 5.27953214364280043565602192295, 5.89868775441805967152972500399, 6.52020618069040882791615760395, 7.42671519653861750834471841347, 7.959200135926492196708824417545, 8.719581538114279246873898705476, 9.396131525526349774390676258206

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