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Properties

Label 2-55e2-55.53-c0-0-0
Degree $2$
Conductor $3025$
Sign $-0.399 + 0.916i$
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 + 1.26i)2-s + (−0.587 + 0.809i)4-s + (−1.39 + 0.221i)7-s + (−0.951 − 0.309i)9-s + (−1.26 + 0.642i)13-s + (−1.17 − 1.61i)14-s + (0.309 + 0.951i)16-s + (−1.26 − 0.642i)17-s + (−0.221 − 1.39i)18-s + (−1.61 − 1.17i)26-s + (0.642 − 1.26i)28-s + (−1 + 1.00i)32-s − 2.00i·34-s + (0.809 − 0.587i)36-s + (−1 − i)43-s + ⋯
L(s)  = 1  + (0.642 + 1.26i)2-s + (−0.587 + 0.809i)4-s + (−1.39 + 0.221i)7-s + (−0.951 − 0.309i)9-s + (−1.26 + 0.642i)13-s + (−1.17 − 1.61i)14-s + (0.309 + 0.951i)16-s + (−1.26 − 0.642i)17-s + (−0.221 − 1.39i)18-s + (−1.61 − 1.17i)26-s + (0.642 − 1.26i)28-s + (−1 + 1.00i)32-s − 2.00i·34-s + (0.809 − 0.587i)36-s + (−1 − i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $-0.399 + 0.916i$
Analytic conductor: \(1.50967\)
Root analytic conductor: \(1.22868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :0),\ -0.399 + 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3373002104\)
\(L(\frac12)\) \(\approx\) \(0.3373002104\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.642 - 1.26i)T + (-0.587 + 0.809i)T^{2} \)
3 \( 1 + (0.951 + 0.309i)T^{2} \)
7 \( 1 + (1.39 - 0.221i)T + (0.951 - 0.309i)T^{2} \)
13 \( 1 + (1.26 - 0.642i)T + (0.587 - 0.809i)T^{2} \)
17 \( 1 + (1.26 + 0.642i)T + (0.587 + 0.809i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.951 - 0.309i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 + (-0.951 - 0.309i)T^{2} \)
53 \( 1 + (0.587 - 0.809i)T^{2} \)
59 \( 1 + (1.17 - 1.61i)T + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + (0.618 + 1.90i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.221 - 1.39i)T + (-0.951 + 0.309i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.642 - 1.26i)T + (-0.587 - 0.809i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.587 + 0.809i)T^{2} \)
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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

−9.165450820803842473172931510246, −8.670535336021774646294537287370, −7.60429551757672113892104912689, −6.87243845272856433136354966549, −6.50409881714103014519174346134, −5.74371404178640591637540101704, −4.99167220424693416281115324922, −4.19175377742996514270301260908, −3.16858065542030548570397756231, −2.29395783949031322591707468864, 0.14815989842318793156768181284, 1.95396150661223116785267095943, 2.85301506212414657086195051423, 3.27107954638736288482267864774, 4.31778832530003591221835543417, 5.04808804802071056446953520760, 5.97284361847846676273050589678, 6.75850226630029728097478131482, 7.64031668209273804601562270473, 8.543356926764619306937316478289

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