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Properties

Label 2-55e2-275.212-c0-0-0
Degree $2$
Conductor $3025$
Sign $-0.0561 + 0.998i$
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  + (0.0489 − 0.309i)3-s − i·4-s + (0.587 − 0.809i)5-s + (0.857 + 0.278i)9-s + (−0.309 − 0.0489i)12-s + (−0.221 − 0.221i)15-s − 16-s + (−0.809 − 0.587i)20-s + (1.76 − 0.278i)23-s + (−0.309 − 0.951i)25-s + (0.270 − 0.530i)27-s + (0.363 + 1.11i)31-s + (0.278 − 0.857i)36-s + (0.642 − 1.26i)37-s + (0.729 − 0.530i)45-s + ⋯
L(s)  = 1  + (0.0489 − 0.309i)3-s − i·4-s + (0.587 − 0.809i)5-s + (0.857 + 0.278i)9-s + (−0.309 − 0.0489i)12-s + (−0.221 − 0.221i)15-s − 16-s + (−0.809 − 0.587i)20-s + (1.76 − 0.278i)23-s + (−0.309 − 0.951i)25-s + (0.270 − 0.530i)27-s + (0.363 + 1.11i)31-s + (0.278 − 0.857i)36-s + (0.642 − 1.26i)37-s + (0.729 − 0.530i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0561 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0561 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $-0.0561 + 0.998i$
Analytic conductor: \(1.50967\)
Root analytic conductor: \(1.22868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :0),\ -0.0561 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.518178191\)
\(L(\frac12)\) \(\approx\) \(1.518178191\)
\(L(1)\) 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 \( 1 + iT^{2} \)
3 \( 1 + (-0.0489 + 0.309i)T + (-0.951 - 0.309i)T^{2} \)
7 \( 1 + (0.587 - 0.809i)T^{2} \)
13 \( 1 + (-0.587 + 0.809i)T^{2} \)
17 \( 1 + (0.951 + 0.309i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-1.76 + 0.278i)T + (0.951 - 0.309i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.642 + 1.26i)T + (-0.587 - 0.809i)T^{2} \)
41 \( 1 + (0.309 + 0.951i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (1.39 + 0.221i)T + (0.951 + 0.309i)T^{2} \)
53 \( 1 + (1.95 - 0.309i)T + (0.951 - 0.309i)T^{2} \)
59 \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.809 - 0.412i)T + (0.587 - 0.809i)T^{2} \)
71 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.951 - 0.309i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.951 - 0.309i)T^{2} \)
89 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (0.309 + 1.95i)T + (-0.951 + 0.309i)T^{2} \)
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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

−8.866653445834673161863467958335, −8.027128791308354684969023185764, −7.01280810700940145374281591891, −6.48988729923740968719223863310, −5.56625751255208882681444458394, −4.91146215251029909467650071508, −4.33768890705519631908100225012, −2.81684631236026868038911719860, −1.71678648888658086854541707595, −1.05578638127609028939987590595, 1.63960939484189177820789198160, 2.86749145013432319844365120619, 3.36054700628783597375553621519, 4.36682111586622409879039389927, 5.10351337031143697561375676164, 6.37550562625922862321542919563, 6.80381449984940814359595349049, 7.57408055754067600452530362692, 8.258572608115218822618016205939, 9.339847308453376596140281288907

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