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Properties

Label 2-2100-7.4-c1-0-20
Degree $2$
Conductor $2100$
Sign $-0.0633 + 0.997i$
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  + (−0.5 + 0.866i)3-s + (1.32 − 2.29i)7-s + (−0.499 − 0.866i)9-s + (0.177 − 0.306i)11-s − 13-s + (1.82 − 3.15i)17-s + (−0.322 − 0.559i)19-s + (1.32 + 2.29i)21-s + (−1 − 1.73i)23-s + 0.999·27-s − 3.29·29-s + (−2 + 3.46i)31-s + (0.177 + 0.306i)33-s + (−1.14 − 1.98i)37-s + (0.5 − 0.866i)39-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.499 − 0.866i)7-s + (−0.166 − 0.288i)9-s + (0.0534 − 0.0925i)11-s − 0.277·13-s + (0.442 − 0.765i)17-s + (−0.0740 − 0.128i)19-s + (0.288 + 0.499i)21-s + (−0.208 − 0.361i)23-s + 0.192·27-s − 0.611·29-s + (−0.359 + 0.622i)31-s + (0.0308 + 0.0534i)33-s + (−0.188 − 0.326i)37-s + (0.0800 − 0.138i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.0633 + 0.997i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.0633 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.094687326\)
\(L(\frac12)\) \(\approx\) \(1.094687326\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 \)
7 \( 1 + (-1.32 + 2.29i)T \)
good11 \( 1 + (-0.177 + 0.306i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (-1.82 + 3.15i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.322 + 0.559i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.29T + 29T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.14 + 1.98i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.64T + 41T^{2} \)
43 \( 1 + 3.29T + 43T^{2} \)
47 \( 1 + (6.46 + 11.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.82 + 6.62i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.468 + 0.811i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.79 - 10.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.96 + 6.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.29T + 71T^{2} \)
73 \( 1 + (-2.14 + 3.71i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.677 - 1.17i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.64T + 83T^{2} \)
89 \( 1 + (3.46 + 6.00i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.70T + 97T^{2} \)
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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

−8.894165583230506450504770602923, −8.180855284243402403465425896421, −7.23996046771952952315750066326, −6.70687128231397452117424563921, −5.47886809451890061693728663487, −4.93352777486883076971376418313, −4.01290068718196905377431338845, −3.21087560518290963776558188442, −1.81440214378149941329638288945, −0.40180186524654997485315870058, 1.42745328190714555448014978941, 2.29075832846758112812193804957, 3.44919644329299652648446655580, 4.60012284384306102837471553770, 5.48301074512506970548085492341, 6.04927993799897177030509708446, 6.96536158493199129487044825318, 7.87541352334291977191132751531, 8.346148937779003071467517044005, 9.282646362697000895823726870508

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