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Properties

Label 2-55e2-55.27-c0-0-1
Degree $2$
Conductor $3025$
Sign $0.916 - 0.399i$
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  + (1.39 − 0.221i)3-s + (0.587 + 0.809i)4-s + (0.951 − 0.309i)9-s + (1 + i)12-s + (−0.309 + 0.951i)16-s + (1 − i)23-s + (0.809 + 0.587i)36-s + (−1.39 − 0.221i)37-s + (0.221 + 1.39i)47-s + (−0.221 + 1.39i)48-s + (−0.951 − 0.309i)49-s + (1.26 + 0.642i)53-s + (−0.951 + 0.309i)64-s + (−1 − i)67-s + (1.17 − 1.61i)69-s + ⋯
L(s)  = 1  + (1.39 − 0.221i)3-s + (0.587 + 0.809i)4-s + (0.951 − 0.309i)9-s + (1 + i)12-s + (−0.309 + 0.951i)16-s + (1 − i)23-s + (0.809 + 0.587i)36-s + (−1.39 − 0.221i)37-s + (0.221 + 1.39i)47-s + (−0.221 + 1.39i)48-s + (−0.951 − 0.309i)49-s + (1.26 + 0.642i)53-s + (−0.951 + 0.309i)64-s + (−1 − i)67-s + (1.17 − 1.61i)69-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $0.916 - 0.399i$
Analytic conductor: \(1.50967\)
Root analytic conductor: \(1.22868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (632, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :0),\ 0.916 - 0.399i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.270476253\)
\(L(\frac12)\) \(\approx\) \(2.270476253\)
\(L(1)\) 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 \( 1 + (-0.587 - 0.809i)T^{2} \)
3 \( 1 + (-1.39 + 0.221i)T + (0.951 - 0.309i)T^{2} \)
7 \( 1 + (0.951 + 0.309i)T^{2} \)
13 \( 1 + (0.587 + 0.809i)T^{2} \)
17 \( 1 + (0.587 - 0.809i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (-1 + i)T - iT^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (1.39 + 0.221i)T + (0.951 + 0.309i)T^{2} \)
41 \( 1 + (0.309 + 0.951i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.221 - 1.39i)T + (-0.951 + 0.309i)T^{2} \)
53 \( 1 + (-1.26 - 0.642i)T + (0.587 + 0.809i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (1 + i)T + iT^{2} \)
71 \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.951 - 0.309i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.587 + 0.809i)T^{2} \)
89 \( 1 + 2iT - T^{2} \)
97 \( 1 + (0.642 - 1.26i)T + (-0.587 - 0.809i)T^{2} \)
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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

−8.863751765997085951189946704074, −8.179789865753001288666554822871, −7.58934316250327788740053821705, −6.96166825427562101568077041653, −6.20213106743448742448519638402, −4.91492012298908588042011361447, −3.93027754472972759462463572453, −3.16644566711627093041461194584, −2.57821892992734463337175295318, −1.66391820586550627636415745115, 1.40013782946010643756124296527, 2.32098942418055076771016752300, 3.12091427679614576616220705501, 3.92139335555234560804161980405, 5.06842551655783928328380166710, 5.71547438554410451377073818146, 6.89340546997023570581657646605, 7.25389009957670675520643720412, 8.291852635883738424305330469172, 8.819168133081378040171724295896

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