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Properties

Label 2-2100-35.4-c1-0-14
Degree $2$
Conductor $2100$
Sign $0.610 + 0.792i$
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 + (2.09 − 1.62i)7-s + (0.499 + 0.866i)9-s + (2.12 − 3.67i)11-s + 5i·13-s + (−3.67 − 2.12i)17-s + (1.62 + 2.80i)19-s + (−2.62 + 0.358i)21-s + (5.19 − 3i)23-s − 0.999i·27-s + 8.48·29-s + (2 − 3.46i)31-s + (−3.67 + 2.12i)33-s + (−4.33 + 2.5i)37-s + (2.5 − 4.33i)39-s + ⋯
L(s)  = 1  + (−0.499 − 0.288i)3-s + (0.790 − 0.612i)7-s + (0.166 + 0.288i)9-s + (0.639 − 1.10i)11-s + 1.38i·13-s + (−0.891 − 0.514i)17-s + (0.371 + 0.644i)19-s + (−0.572 + 0.0782i)21-s + (1.08 − 0.625i)23-s − 0.192i·27-s + 1.57·29-s + (0.359 − 0.622i)31-s + (−0.639 + 0.369i)33-s + (−0.711 + 0.410i)37-s + (0.400 − 0.693i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.610 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.610 + 0.792i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.610 + 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.684794680\)
\(L(\frac12)\) \(\approx\) \(1.684794680\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.09 + 1.62i)T \)
good11 \( 1 + (-2.12 + 3.67i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5iT - 13T^{2} \)
17 \( 1 + (3.67 + 2.12i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.62 - 2.80i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.19 + 3i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.48T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.33 - 2.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.75T + 41T^{2} \)
43 \( 1 - 4.48iT - 43T^{2} \)
47 \( 1 + (1.52 - 0.878i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.67 - 2.12i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.878 + 1.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.8 + 6.86i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.48T + 71T^{2} \)
73 \( 1 + (11.6 + 6.74i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.378 - 0.655i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.2iT - 83T^{2} \)
89 \( 1 + (-8.12 - 14.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.9iT - 97T^{2} \)
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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

−8.806590466539987507523466537213, −8.345431733602794730060982603811, −7.24457551679357425347943179197, −6.67835378124183581394635400597, −5.98586241168300251499828302705, −4.77421443935686819407774780879, −4.36393529522113494801634810576, −3.13297758307730287525237891984, −1.78581031836153259493897039887, −0.794768379111883986325875775766, 1.09873134583288621434979344971, 2.32025056911417423151340820081, 3.43985344424925965053567064032, 4.69860632061251136337954291184, 5.01406679808795974684263565447, 5.98552662760762107296648273232, 6.88620511643709330282375739844, 7.58129516307204330532026888246, 8.675060684262346667362469702994, 9.025970185862425472525616630992

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