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Properties

Label 2-2100-35.9-c1-0-4
Degree $2$
Conductor $2100$
Sign $-0.123 - 0.992i$
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  + (−0.866 + 0.5i)3-s + (−2.29 − 1.32i)7-s + (0.499 − 0.866i)9-s + (0.822 + 1.42i)11-s + 2.64i·13-s + (−1.42 + 0.822i)17-s + (4.14 − 7.18i)19-s + 2.64·21-s + (1.42 + 0.822i)23-s + 0.999i·27-s − 7.64·29-s + (2.14 + 3.71i)31-s + (−1.42 − 0.822i)33-s + (−0.559 − 0.322i)37-s + (−1.32 − 2.29i)39-s + ⋯
L(s)  = 1  + (−0.499 + 0.288i)3-s + (−0.866 − 0.499i)7-s + (0.166 − 0.288i)9-s + (0.248 + 0.429i)11-s + 0.733i·13-s + (−0.345 + 0.199i)17-s + (0.951 − 1.64i)19-s + 0.577·21-s + (0.297 + 0.171i)23-s + 0.192i·27-s − 1.41·29-s + (0.385 + 0.667i)31-s + (−0.248 − 0.143i)33-s + (−0.0919 − 0.0530i)37-s + (−0.211 − 0.366i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.123 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.123 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.123 - 0.992i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.123 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8770419567\)
\(L(\frac12)\) \(\approx\) \(0.8770419567\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
7 \( 1 + (2.29 + 1.32i)T \)
good11 \( 1 + (-0.822 - 1.42i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.64iT - 13T^{2} \)
17 \( 1 + (1.42 - 0.822i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.14 + 7.18i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.42 - 0.822i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.64T + 29T^{2} \)
31 \( 1 + (-2.14 - 3.71i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.559 + 0.322i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.93T + 41T^{2} \)
43 \( 1 - 5.93iT - 43T^{2} \)
47 \( 1 + (5.19 + 3i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.85 - 1.64i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.46 - 9.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.559 + 0.322i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + (11.4 - 6.61i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.14 + 1.98i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.9iT - 83T^{2} \)
89 \( 1 + (7.11 - 12.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−9.421479447285227764033232896531, −8.820575819529908908249811745950, −7.45884982632252398829251811073, −6.95394346898707860958609958258, −6.28645767568046472676021102219, −5.26803665095405792400545040459, −4.45697065034218007780297751342, −3.65696374562989693414014619802, −2.60043835360970783003067927992, −1.09935860618127219007856242167, 0.37954326800083424113225812708, 1.81631873583150509647258055526, 3.07603451184552925529554082822, 3.80566009899217009417076343302, 5.12964194614714988610414921105, 5.84225920419664343222361752427, 6.33541652410325893913534653468, 7.36437924966998761787848790491, 8.010178991920593573241107057699, 8.958699561803741134240566015115

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