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Properties

Label 2-55e2-275.103-c0-0-0
Degree $2$
Conductor $3025$
Sign $0.689 - 0.724i$
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  + (0.896 + 0.142i)3-s + (0.951 + 0.309i)4-s + (0.587 + 0.809i)5-s + (−0.166 − 0.0542i)9-s + (0.809 + 0.412i)12-s + (0.412 + 0.809i)15-s + (0.809 + 0.587i)16-s + (0.309 + 0.951i)20-s + (0.278 − 0.142i)23-s + (−0.309 + 0.951i)25-s + (−0.951 − 0.484i)27-s − 1.90·31-s + (−0.142 − 0.103i)36-s + (0.221 − 1.39i)37-s + (−0.0542 − 0.166i)45-s + ⋯
L(s)  = 1  + (0.896 + 0.142i)3-s + (0.951 + 0.309i)4-s + (0.587 + 0.809i)5-s + (−0.166 − 0.0542i)9-s + (0.809 + 0.412i)12-s + (0.412 + 0.809i)15-s + (0.809 + 0.587i)16-s + (0.309 + 0.951i)20-s + (0.278 − 0.142i)23-s + (−0.309 + 0.951i)25-s + (−0.951 − 0.484i)27-s − 1.90·31-s + (−0.142 − 0.103i)36-s + (0.221 − 1.39i)37-s + (−0.0542 − 0.166i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $0.689 - 0.724i$
Analytic conductor: \(1.50967\)
Root analytic conductor: \(1.22868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (928, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :0),\ 0.689 - 0.724i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.219310866\)
\(L(\frac12)\) \(\approx\) \(2.219310866\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.587 - 0.809i)T \)
11 \( 1 \)
good2 \( 1 + (-0.951 - 0.309i)T^{2} \)
3 \( 1 + (-0.896 - 0.142i)T + (0.951 + 0.309i)T^{2} \)
7 \( 1 + (0.587 + 0.809i)T^{2} \)
13 \( 1 + (0.587 - 0.809i)T^{2} \)
17 \( 1 + (-0.587 + 0.809i)T^{2} \)
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 \( 1 + 1.90T + T^{2} \)
37 \( 1 + (-0.221 + 1.39i)T + (-0.951 - 0.309i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.221 - 1.39i)T + (-0.951 - 0.309i)T^{2} \)
53 \( 1 + (-0.809 + 1.58i)T + (-0.587 - 0.809i)T^{2} \)
59 \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.309 + 1.95i)T + (-0.951 - 0.309i)T^{2} \)
71 \( 1 - 0.618T + T^{2} \)
73 \( 1 + iT^{2} \)
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 \( 1 + (1.58 + 0.809i)T + (0.587 + 0.809i)T^{2} \)
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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

−9.012601287904304952388262355884, −8.205887599375136363839988152113, −7.43201575884234664936540848668, −6.91447214601253256559502481054, −6.04024781909179320296318348341, −5.40650227876087600848049280240, −3.86875195280195535636325971538, −3.28722024088792694026511042561, −2.47368658231474662610041889625, −1.85320374243498700534254792946, 1.35364500982676398930349298139, 2.16575676738601209133418990327, 2.93712937216136342141289722213, 3.93296716953408260515976340643, 5.21714169273667737901865116863, 5.65413465375447136797971111242, 6.62153403313566874353934585010, 7.35689124383917970747574966547, 8.168452894237036493223491184465, 8.729317477527585569819018133465

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