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Properties

Label 2-55e2-275.108-c0-0-0
Degree $2$
Conductor $3025$
Sign $0.962 + 0.269i$
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  + (1.58 − 0.809i)3-s + i·4-s + (0.951 − 0.309i)5-s + (1.27 − 1.76i)9-s + (0.809 + 1.58i)12-s + (1.26 − 1.26i)15-s − 16-s + (0.309 + 0.951i)20-s + (−0.896 + 1.76i)23-s + (0.809 − 0.587i)25-s + (0.327 − 2.06i)27-s + (−1.53 + 1.11i)31-s + (1.76 + 1.27i)36-s + (0.221 − 1.39i)37-s + (0.672 − 2.06i)45-s + ⋯
L(s)  = 1  + (1.58 − 0.809i)3-s + i·4-s + (0.951 − 0.309i)5-s + (1.27 − 1.76i)9-s + (0.809 + 1.58i)12-s + (1.26 − 1.26i)15-s − 16-s + (0.309 + 0.951i)20-s + (−0.896 + 1.76i)23-s + (0.809 − 0.587i)25-s + (0.327 − 2.06i)27-s + (−1.53 + 1.11i)31-s + (1.76 + 1.27i)36-s + (0.221 − 1.39i)37-s + (0.672 − 2.06i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $0.962 + 0.269i$
Analytic conductor: \(1.50967\)
Root analytic conductor: \(1.22868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (2308, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :0),\ 0.962 + 0.269i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.454115321\)
\(L(\frac12)\) \(\approx\) \(2.454115321\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 \)
good2 \( 1 - iT^{2} \)
3 \( 1 + (-1.58 + 0.809i)T + (0.587 - 0.809i)T^{2} \)
7 \( 1 + (0.951 - 0.309i)T^{2} \)
13 \( 1 + (-0.951 + 0.309i)T^{2} \)
17 \( 1 + (-0.587 + 0.809i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.896 - 1.76i)T + (-0.587 - 0.809i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.221 + 1.39i)T + (-0.951 - 0.309i)T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.642 + 1.26i)T + (-0.587 + 0.809i)T^{2} \)
53 \( 1 + (0.412 - 0.809i)T + (-0.587 - 0.809i)T^{2} \)
59 \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.309 - 0.951i)T^{2} \)
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 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.587 - 0.809i)T^{2} \)
89 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (-0.809 - 0.412i)T + (0.587 + 0.809i)T^{2} \)
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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

−9.007677808372735851106674541642, −8.028456345254898367958465395294, −7.59276923616103017665654755261, −6.91210275579785491590206861748, −6.03875999129412472774435344409, −4.92494797903440923530125561020, −3.67380528645201677822199213626, −3.28213355002778604457683430291, −2.16076486784676587569108267998, −1.67328947559733140363553087774, 1.68472798265226081593725377658, 2.36282004023315052485296034160, 3.17006240232383890891631146132, 4.30173851838700696008381115564, 4.89212714294335021254944155239, 5.92292231025915578067949711032, 6.57473974128248327735995555416, 7.60873144073075980203765681441, 8.435140636372141237613607900287, 9.117685930092708413657535101031

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