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Properties

Label 2-55e2-275.192-c0-0-0
Degree $2$
Conductor $3025$
Sign $-0.0847 - 0.996i$
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  + (−0.642 + 0.642i)3-s + (−0.951 − 0.309i)4-s + (0.951 + 0.309i)5-s + 0.175i·9-s + (0.809 − 0.412i)12-s + (−0.809 + 0.412i)15-s + (0.809 + 0.587i)16-s + (−0.809 − 0.587i)20-s + (0.278 + 0.142i)23-s + (0.809 + 0.587i)25-s + (−0.754 − 0.754i)27-s + (−0.587 + 1.80i)31-s + (0.0542 − 0.166i)36-s + (0.642 − 1.26i)37-s + (−0.0542 + 0.166i)45-s + ⋯
L(s)  = 1  + (−0.642 + 0.642i)3-s + (−0.951 − 0.309i)4-s + (0.951 + 0.309i)5-s + 0.175i·9-s + (0.809 − 0.412i)12-s + (−0.809 + 0.412i)15-s + (0.809 + 0.587i)16-s + (−0.809 − 0.587i)20-s + (0.278 + 0.142i)23-s + (0.809 + 0.587i)25-s + (−0.754 − 0.754i)27-s + (−0.587 + 1.80i)31-s + (0.0542 − 0.166i)36-s + (0.642 − 1.26i)37-s + (−0.0542 + 0.166i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0847 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0847 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $-0.0847 - 0.996i$
Analytic conductor: \(1.50967\)
Root analytic conductor: \(1.22868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (1842, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :0),\ -0.0847 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8388732526\)
\(L(\frac12)\) \(\approx\) \(0.8388732526\)
\(L(1)\) 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 \( 1 + (0.951 + 0.309i)T^{2} \)
3 \( 1 + (0.642 - 0.642i)T - iT^{2} \)
7 \( 1 + (0.951 + 0.309i)T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (0.951 + 0.309i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.278 - 0.142i)T + (0.587 + 0.809i)T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.642 + 1.26i)T + (-0.587 - 0.809i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-1 - i)T + iT^{2} \)
53 \( 1 + (-1.76 + 0.278i)T + (0.951 - 0.309i)T^{2} \)
59 \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (-0.309 - 1.95i)T + (-0.951 + 0.309i)T^{2} \)
71 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.951 - 0.309i)T^{2} \)
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 \( 1 + (-0.278 - 1.76i)T + (-0.951 + 0.309i)T^{2} \)
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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

−9.240511991517470994533327291559, −8.640077761614336477735171796591, −7.59488598986940116035401087760, −6.66890897669004725565147244430, −5.68225300823042213725417736309, −5.40838099622360754001887549102, −4.60448542754098619432148171060, −3.78721203060416830183468401064, −2.59022793450989999918233423113, −1.31872211638234270028024220250, 0.64331355861227577311743559581, 1.79046973332945852553137445264, 3.04336517406122522245283100152, 4.13816115922653864419984681099, 4.96034211284698554072316279585, 5.74370275243680161169417258751, 6.25348854905423026899893726065, 7.17366781802514365298363315091, 7.975689292163218222289370782774, 8.795408057454783611203500054326

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