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Properties

Label 2-2100-15.2-c1-0-1
Degree $2$
Conductor $2100$
Sign $-0.621 + 0.783i$
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  + (−0.573 + 1.63i)3-s + (−0.707 − 0.707i)7-s + (−2.34 − 1.87i)9-s + 6.10i·11-s + (−2.13 + 2.13i)13-s + (−1.21 + 1.21i)17-s − 0.390i·19-s + (1.56 − 0.750i)21-s + (−2.04 − 2.04i)23-s + (4.40 − 2.75i)27-s + 5.67·29-s + 1.37·31-s + (−9.97 − 3.50i)33-s + (−2.84 − 2.84i)37-s + (−2.26 − 4.71i)39-s + ⋯
L(s)  = 1  + (−0.331 + 0.943i)3-s + (−0.267 − 0.267i)7-s + (−0.780 − 0.624i)9-s + 1.84i·11-s + (−0.591 + 0.591i)13-s + (−0.294 + 0.294i)17-s − 0.0895i·19-s + (0.340 − 0.163i)21-s + (−0.426 − 0.426i)23-s + (0.848 − 0.529i)27-s + 1.05·29-s + 0.246·31-s + (−1.73 − 0.609i)33-s + (−0.467 − 0.467i)37-s + (−0.362 − 0.754i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.621 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.621 + 0.783i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.621 + 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2094764698\)
\(L(\frac12)\) \(\approx\) \(0.2094764698\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.573 - 1.63i)T \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good11 \( 1 - 6.10iT - 11T^{2} \)
13 \( 1 + (2.13 - 2.13i)T - 13iT^{2} \)
17 \( 1 + (1.21 - 1.21i)T - 17iT^{2} \)
19 \( 1 + 0.390iT - 19T^{2} \)
23 \( 1 + (2.04 + 2.04i)T + 23iT^{2} \)
29 \( 1 - 5.67T + 29T^{2} \)
31 \( 1 - 1.37T + 31T^{2} \)
37 \( 1 + (2.84 + 2.84i)T + 37iT^{2} \)
41 \( 1 + 5.59iT - 41T^{2} \)
43 \( 1 + (3.04 - 3.04i)T - 43iT^{2} \)
47 \( 1 + (1.32 - 1.32i)T - 47iT^{2} \)
53 \( 1 + (9.10 + 9.10i)T + 53iT^{2} \)
59 \( 1 - 6.21T + 59T^{2} \)
61 \( 1 + 6.30T + 61T^{2} \)
67 \( 1 + (7.63 + 7.63i)T + 67iT^{2} \)
71 \( 1 - 6.26iT - 71T^{2} \)
73 \( 1 + (11.3 - 11.3i)T - 73iT^{2} \)
79 \( 1 + 6.56iT - 79T^{2} \)
83 \( 1 + (-3.59 - 3.59i)T + 83iT^{2} \)
89 \( 1 + 17.7T + 89T^{2} \)
97 \( 1 + (10.3 + 10.3i)T + 97iT^{2} \)
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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

−9.824552898204898285186880235915, −9.038025972355308309319795459893, −8.148642623625340645736937047763, −7.05571582001994253918412235944, −6.59336295844098359576967690781, −5.48305415304246261803749609013, −4.50331481356680189675964286969, −4.28724943593558337957984645800, −2.98726437129788559767157906243, −1.87359328409952596881620038045, 0.078900597634937142905162043701, 1.26145073458327562730885696690, 2.68254893025664458764109036747, 3.26508141357217448320823066997, 4.75157452197319478406438822677, 5.67646482418729766682451929494, 6.16959661306064769316838423292, 6.97129001942694178802047303021, 7.940843982212369788522845436057, 8.396072547184806627397802235259

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