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Properties

Label 2-55e2-11.8-c0-0-0
Degree $2$
Conductor $3025$
Sign $0.286 + 0.958i$
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 − 0.951i)4-s + (0.809 − 0.587i)9-s + (−0.809 − 0.587i)16-s + (1.61 − 1.17i)31-s + (−0.309 − 0.951i)36-s + (−0.809 − 0.587i)49-s + (−0.618 + 1.90i)59-s + (−0.809 + 0.587i)64-s + (−1.61 − 1.17i)71-s + (0.309 − 0.951i)81-s + 2·89-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)4-s + (0.809 − 0.587i)9-s + (−0.809 − 0.587i)16-s + (1.61 − 1.17i)31-s + (−0.309 − 0.951i)36-s + (−0.809 − 0.587i)49-s + (−0.618 + 1.90i)59-s + (−0.809 + 0.587i)64-s + (−1.61 − 1.17i)71-s + (0.309 − 0.951i)81-s + 2·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $0.286 + 0.958i$
Analytic conductor: \(1.50967\)
Root analytic conductor: \(1.22868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (2901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :0),\ 0.286 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.389961530\)
\(L(\frac12)\) \(\approx\) \(1.389961530\)
\(L(1)\) 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 \( 1 + (-0.309 + 0.951i)T^{2} \)
3 \( 1 + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-1.61 + 1.17i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.618 - 1.90i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 + (0.309 - 0.951i)T^{2} \)
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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

−8.948599195229487476247271131885, −7.903078198723099393824615954269, −7.14909144222917486213540542846, −6.38688192114351780713308688419, −5.89237501676137510682339954913, −4.83279588719347990127203577688, −4.20946782737479113767621070635, −3.02747941243955528740200309123, −1.95125099815500254925731718884, −0.921513808326219850436467154319, 1.54933882242620815637663757873, 2.60517708748495607011610822641, 3.44184515103002527813404887507, 4.40127496899301648621571448866, 4.99569410438809019900586885819, 6.26215594049809338450300095306, 6.87372306665595397042354284148, 7.63654910972502878883701996790, 8.175375862722574562854153422966, 8.907463579617866558483013650831

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