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Properties

Label 2-2100-35.9-c1-0-18
Degree $2$
Conductor $2100$
Sign $0.502 + 0.864i$
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 + (0.177 + 0.306i)11-s − i·13-s + (3.15 − 1.82i)17-s + (0.322 − 0.559i)19-s + (1.32 − 2.29i)21-s + (−1.73 − i)23-s − 0.999i·27-s + 3.29·29-s + (−2 − 3.46i)31-s + (0.306 + 0.177i)33-s + (1.98 + 1.14i)37-s + (−0.5 − 0.866i)39-s + ⋯
L(s)  = 1  + (0.499 − 0.288i)3-s + (0.866 − 0.499i)7-s + (0.166 − 0.288i)9-s + (0.0534 + 0.0925i)11-s − 0.277i·13-s + (0.765 − 0.442i)17-s + (0.0740 − 0.128i)19-s + (0.288 − 0.499i)21-s + (−0.361 − 0.208i)23-s − 0.192i·27-s + 0.611·29-s + (−0.359 − 0.622i)31-s + (0.0534 + 0.0308i)33-s + (0.326 + 0.188i)37-s + (−0.0800 − 0.138i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.502 + 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.502 + 0.864i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.502 + 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.371621617\)
\(L(\frac12)\) \(\approx\) \(2.371621617\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.177 - 0.306i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 + (-3.15 + 1.82i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.322 + 0.559i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.73 + i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.29T + 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.98 - 1.14i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.64T + 41T^{2} \)
43 \( 1 + 3.29iT - 43T^{2} \)
47 \( 1 + (-11.2 - 6.46i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.62 - 3.82i)T + (26.5 - 45.8i)T^{2} \)
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 \( 1 + (-6.87 + 3.96i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.29T + 71T^{2} \)
73 \( 1 + (3.71 - 2.14i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.677 - 1.17i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.64iT - 83T^{2} \)
89 \( 1 + (-3.46 + 6.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.70iT - 97T^{2} \)
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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

−8.879324349114311264900128282122, −8.052815174625356737268927085310, −7.60200557886506508268723670866, −6.80833337299318368173600873304, −5.79230392192235584285552226093, −4.88502803522918942853700361095, −4.04622400679354806633621436393, −3.05480444276779966103360335927, −1.98489142817926962759222642293, −0.864770647640092267004921211764, 1.37832075107405269605979412158, 2.38022758298277013612910741321, 3.44414164245018148193369438060, 4.33232065919120012610987984450, 5.21796316437154512002665353983, 5.93330529653060374434330832775, 7.04226587547136381665872128633, 7.85479034571900158970262998546, 8.497951195587259192804474321937, 9.071815275831113063831749134329

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