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Properties

Label 2-55e2-275.148-c0-0-0
Degree $2$
Conductor $3025$
Sign $0.462 - 0.886i$
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  + (0.309 + 0.0489i)3-s + (−0.951 + 0.309i)4-s + (−0.951 − 0.309i)5-s + (−0.857 − 0.278i)9-s + (−0.309 + 0.0489i)12-s + (−0.278 − 0.142i)15-s + (0.809 − 0.587i)16-s + 0.999·20-s + (1.76 + 0.278i)23-s + (0.809 + 0.587i)25-s + (−0.530 − 0.270i)27-s + (−0.951 + 0.690i)31-s + 0.902·36-s + (−1 + i)37-s + (0.729 + 0.530i)45-s + ⋯
L(s)  = 1  + (0.309 + 0.0489i)3-s + (−0.951 + 0.309i)4-s + (−0.951 − 0.309i)5-s + (−0.857 − 0.278i)9-s + (−0.309 + 0.0489i)12-s + (−0.278 − 0.142i)15-s + (0.809 − 0.587i)16-s + 0.999·20-s + (1.76 + 0.278i)23-s + (0.809 + 0.587i)25-s + (−0.530 − 0.270i)27-s + (−0.951 + 0.690i)31-s + 0.902·36-s + (−1 + i)37-s + (0.729 + 0.530i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $0.462 - 0.886i$
Analytic conductor: \(1.50967\)
Root analytic conductor: \(1.22868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :0),\ 0.462 - 0.886i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6786205221\)
\(L(\frac12)\) \(\approx\) \(0.6786205221\)
\(L(1)\) 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 \( 1 + (-0.309 - 0.0489i)T + (0.951 + 0.309i)T^{2} \)
7 \( 1 + (-0.951 - 0.309i)T^{2} \)
13 \( 1 + (0.587 - 0.809i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (-1.76 - 0.278i)T + (0.951 + 0.309i)T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (1 - i)T - iT^{2} \)
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 \( 1 + (-1.39 - 1.39i)T + iT^{2} \)
59 \( 1 - 1.17iT - T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.809 + 0.412i)T + (0.587 + 0.809i)T^{2} \)
71 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.587 - 0.809i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (-1.39 + 1.39i)T - iT^{2} \)
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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

−8.770291447506973112424798773095, −8.612932053277941547273863126637, −7.60120830036496282307877276992, −7.08876019309289994944535777902, −5.79636332036896905306493849142, −5.04963828315510920860280878090, −4.32539261497386066101276197783, −3.43660686212874195606755919268, −2.91103626923040039916128438160, −1.04502704792082145211135616245, 0.52073290234760121858287477418, 2.22435065885259019349413828459, 3.39170052245956366942782427635, 3.87793194651326496364336451221, 5.02452583286744143692853316898, 5.43589186428972784445501980282, 6.62806672994062407590538260800, 7.40096334034104178265454423162, 8.158964905999657398878978279207, 8.812032920240819368180172100411

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