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Properties

Label 2-2700-15.2-c1-0-5
Degree $2$
Conductor $2700$
Sign $-0.229 - 0.973i$
Analytic cond. $21.5596$
Root an. cond. $4.64323$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.22 + 2.22i)7-s − 5.67i·11-s + (−1.22 + 1.22i)13-s + (−5.67 + 5.67i)17-s + 5.89i·19-s + (−1.27 − 1.27i)23-s + 5.67·29-s − 4.44·31-s + (3.67 + 3.67i)37-s + 8.23i·41-s + (−5.44 + 5.44i)43-s + 2.89i·49-s + (1.27 + 1.27i)53-s + 2.55·59-s − 6.79·61-s + ⋯
L(s)  = 1  + (0.840 + 0.840i)7-s − 1.71i·11-s + (−0.339 + 0.339i)13-s + (−1.37 + 1.37i)17-s + 1.35i·19-s + (−0.266 − 0.266i)23-s + 1.05·29-s − 0.799·31-s + (0.604 + 0.604i)37-s + 1.28i·41-s + (−0.831 + 0.831i)43-s + 0.414i·49-s + (0.175 + 0.175i)53-s + 0.332·59-s − 0.870·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.229 - 0.973i$
Analytic conductor: \(21.5596\)
Root analytic conductor: \(4.64323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :1/2),\ -0.229 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.337377154\)
\(L(\frac12)\) \(\approx\) \(1.337377154\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-2.22 - 2.22i)T + 7iT^{2} \)
11 \( 1 + 5.67iT - 11T^{2} \)
13 \( 1 + (1.22 - 1.22i)T - 13iT^{2} \)
17 \( 1 + (5.67 - 5.67i)T - 17iT^{2} \)
19 \( 1 - 5.89iT - 19T^{2} \)
23 \( 1 + (1.27 + 1.27i)T + 23iT^{2} \)
29 \( 1 - 5.67T + 29T^{2} \)
31 \( 1 + 4.44T + 31T^{2} \)
37 \( 1 + (-3.67 - 3.67i)T + 37iT^{2} \)
41 \( 1 - 8.23iT - 41T^{2} \)
43 \( 1 + (5.44 - 5.44i)T - 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + (-1.27 - 1.27i)T + 53iT^{2} \)
59 \( 1 - 2.55T + 59T^{2} \)
61 \( 1 + 6.79T + 61T^{2} \)
67 \( 1 + (1.32 + 1.32i)T + 67iT^{2} \)
71 \( 1 + 8.23iT - 71T^{2} \)
73 \( 1 + (-3.77 + 3.77i)T - 73iT^{2} \)
79 \( 1 - 15.2iT - 79T^{2} \)
83 \( 1 + (-6.95 - 6.95i)T + 83iT^{2} \)
89 \( 1 + 2.55T + 89T^{2} \)
97 \( 1 + (-7.22 - 7.22i)T + 97iT^{2} \)
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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

−8.782968616339289411346097872923, −8.330930958308011905891309957447, −7.905934022285585315416407829301, −6.40388842411833116121133833918, −6.15842034248164815362288686074, −5.20853170552423488086132360021, −4.34312100884710912999393656060, −3.41127311385079321456567506388, −2.34021886068339083715409630221, −1.39002452399168208244834783035, 0.42612036151036008201186922931, 1.88466286893707593302346849357, 2.64841722849937317509479201611, 4.08025415267711632957531273911, 4.72877117570237719891239191384, 5.15854217797272187889581658877, 6.61706254766342795736979606948, 7.29170802806780385001305931794, 7.51175692583568728018316405992, 8.739782936520532699281068057705

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