Properties

Label 2-55e2-25.8-c0-0-0
Degree $2$
Conductor $3025$
Sign $0.770 - 0.637i$
Analytic cond. $1.50967$
Root an. cond. $1.22868$
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  + (−1.76 − 0.278i)3-s + (−0.587 − 0.809i)4-s i·5-s + (2.06 + 0.672i)9-s + (0.809 + 1.58i)12-s + (−0.278 + 1.76i)15-s + (−0.309 + 0.951i)16-s + (−0.809 + 0.587i)20-s + (−0.896 + 1.76i)23-s − 25-s + (−1.86 − 0.951i)27-s + (−1.53 − 1.11i)31-s + (−0.672 − 2.06i)36-s + (0.642 + 1.26i)37-s + (0.672 − 2.06i)45-s + ⋯
L(s)  = 1  + (−1.76 − 0.278i)3-s + (−0.587 − 0.809i)4-s i·5-s + (2.06 + 0.672i)9-s + (0.809 + 1.58i)12-s + (−0.278 + 1.76i)15-s + (−0.309 + 0.951i)16-s + (−0.809 + 0.587i)20-s + (−0.896 + 1.76i)23-s − 25-s + (−1.86 − 0.951i)27-s + (−1.53 − 1.11i)31-s + (−0.672 − 2.06i)36-s + (0.642 + 1.26i)37-s + (0.672 − 2.06i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $0.770 - 0.637i$
Analytic conductor: \(1.50967\)
Root analytic conductor: \(1.22868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (2058, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :0),\ 0.770 - 0.637i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2719940719\)
\(L(\frac12)\) \(\approx\) \(0.2719940719\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + iT \)
11 \( 1 \)
good2 \( 1 + (0.587 + 0.809i)T^{2} \)
3 \( 1 + (1.76 + 0.278i)T + (0.951 + 0.309i)T^{2} \)
7 \( 1 - iT^{2} \)
13 \( 1 + (0.587 - 0.809i)T^{2} \)
17 \( 1 + (0.951 - 0.309i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.896 - 1.76i)T + (-0.587 - 0.809i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.642 - 1.26i)T + (-0.587 + 0.809i)T^{2} \)
41 \( 1 + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.221 - 1.39i)T + (-0.951 - 0.309i)T^{2} \)
53 \( 1 + (0.896 + 0.142i)T + (0.951 + 0.309i)T^{2} \)
59 \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.309 + 0.0489i)T + (0.951 - 0.309i)T^{2} \)
71 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.587 - 0.809i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.951 + 0.309i)T^{2} \)
89 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.142 - 0.896i)T + (-0.951 - 0.309i)T^{2} \)
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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

−9.365832963821753976131400375495, −8.131832107868771189631242694627, −7.43455888837884140597670903594, −6.35044867067574322905935488639, −5.76041909710252552555465214140, −5.35457363288893522483073629050, −4.59080411962624072542293109971, −3.93126777635726748280071990278, −1.77298566980545758515108797825, −1.04110780867021398618988765919, 0.26411117566004261905901214670, 2.20002980760064627390965898479, 3.54113289849722366102691657110, 4.14529240666061129387896587704, 5.03673122931825501008402965849, 5.72712222942955676236818797209, 6.63694046439896241776499602868, 7.03350361238538120480696471053, 7.933405995564450088085115142799, 8.866373662273002724050902294679

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