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Properties

Label 2-2100-35.27-c1-0-20
Degree $2$
Conductor $2100$
Sign $-0.645 + 0.763i$
Analytic cond. $16.7685$
Root an. cond. $4.09494$
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  + (0.707 + 0.707i)3-s + (−2.12 + 1.57i)7-s + 1.00i·9-s − 2.40·11-s + (−0.697 − 0.697i)13-s + (−3.50 + 3.50i)17-s + 0.306·19-s + (−2.61 − 0.391i)21-s + (2.63 − 2.63i)23-s + (−0.707 + 0.707i)27-s − 4.12i·29-s − 9.14i·31-s + (−1.69 − 1.69i)33-s + (−6.11 − 6.11i)37-s − 0.986i·39-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.803 + 0.594i)7-s + 0.333i·9-s − 0.723·11-s + (−0.193 − 0.193i)13-s + (−0.849 + 0.849i)17-s + 0.0702·19-s + (−0.571 − 0.0853i)21-s + (0.548 − 0.548i)23-s + (−0.136 + 0.136i)27-s − 0.766i·29-s − 1.64i·31-s + (−0.295 − 0.295i)33-s + (−1.00 − 1.00i)37-s − 0.157i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.645 + 0.763i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.645 + 0.763i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2060639863\)
\(L(\frac12)\) \(\approx\) \(0.2060639863\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 \)
7 \( 1 + (2.12 - 1.57i)T \)
good11 \( 1 + 2.40T + 11T^{2} \)
13 \( 1 + (0.697 + 0.697i)T + 13iT^{2} \)
17 \( 1 + (3.50 - 3.50i)T - 17iT^{2} \)
19 \( 1 - 0.306T + 19T^{2} \)
23 \( 1 + (-2.63 + 2.63i)T - 23iT^{2} \)
29 \( 1 + 4.12iT - 29T^{2} \)
31 \( 1 + 9.14iT - 31T^{2} \)
37 \( 1 + (6.11 + 6.11i)T + 37iT^{2} \)
41 \( 1 - 3.06iT - 41T^{2} \)
43 \( 1 + (3.56 - 3.56i)T - 43iT^{2} \)
47 \( 1 + (-0.325 + 0.325i)T - 47iT^{2} \)
53 \( 1 + (-9.00 + 9.00i)T - 53iT^{2} \)
59 \( 1 + 13.0T + 59T^{2} \)
61 \( 1 - 6.43iT - 61T^{2} \)
67 \( 1 + (9.48 + 9.48i)T + 67iT^{2} \)
71 \( 1 + 9.39T + 71T^{2} \)
73 \( 1 + (3.69 + 3.69i)T + 73iT^{2} \)
79 \( 1 - 12.2iT - 79T^{2} \)
83 \( 1 + (-2.89 - 2.89i)T + 83iT^{2} \)
89 \( 1 - 1.75T + 89T^{2} \)
97 \( 1 + (-7.96 + 7.96i)T - 97iT^{2} \)
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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

−8.870224241996317831010634211931, −8.189123908295529424054918944671, −7.35705195327417637029910129120, −6.35923907111691182465349435453, −5.69525201078325128646023627697, −4.71957633301082313492077075103, −3.82622271398925373812008863243, −2.84581999664207717470894236790, −2.10848735652561526438286269385, −0.06512367502982419525269247096, 1.43444032288825731691121828375, 2.77367431263989572301717911605, 3.35670909340788451233918165764, 4.55122232454912835753869713201, 5.36605508020760621938657184218, 6.52234765678782662345618944015, 7.10113702584212228317327398741, 7.62778754276363437659891753375, 8.852116500252426439941533911495, 9.096812100892796564315730091029

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