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Properties

Label 2-2100-35.3-c1-0-20
Degree $2$
Conductor $2100$
Sign $0.939 + 0.342i$
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  + (0.258 + 0.965i)3-s + (2.54 − 0.728i)7-s + (−0.866 + 0.499i)9-s + (2.14 − 3.71i)11-s + (3.22 − 3.22i)13-s + (−3.44 + 0.923i)17-s + (3.13 + 5.42i)19-s + (1.36 + 2.26i)21-s + (0.749 − 2.79i)23-s + (−0.707 − 0.707i)27-s − 2.84i·29-s + (−1.34 − 0.775i)31-s + (4.14 + 1.11i)33-s + (−5.50 − 1.47i)37-s + (3.95 + 2.28i)39-s + ⋯
L(s)  = 1  + (0.149 + 0.557i)3-s + (0.961 − 0.275i)7-s + (−0.288 + 0.166i)9-s + (0.646 − 1.12i)11-s + (0.895 − 0.895i)13-s + (−0.835 + 0.223i)17-s + (0.719 + 1.24i)19-s + (0.297 + 0.494i)21-s + (0.156 − 0.583i)23-s + (−0.136 − 0.136i)27-s − 0.527i·29-s + (−0.241 − 0.139i)31-s + (0.721 + 0.193i)33-s + (−0.905 − 0.242i)37-s + (0.633 + 0.365i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.176695456\)
\(L(\frac12)\) \(\approx\) \(2.176695456\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 \)
7 \( 1 + (-2.54 + 0.728i)T \)
good11 \( 1 + (-2.14 + 3.71i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.22 + 3.22i)T - 13iT^{2} \)
17 \( 1 + (3.44 - 0.923i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-3.13 - 5.42i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.749 + 2.79i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 2.84iT - 29T^{2} \)
31 \( 1 + (1.34 + 0.775i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.50 + 1.47i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 10.7iT - 41T^{2} \)
43 \( 1 + (2.38 + 2.38i)T + 43iT^{2} \)
47 \( 1 + (-2.22 + 8.28i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (8.76 - 2.34i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.820 + 1.42i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.94 + 5.16i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.19 - 4.46i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 + (-2.40 - 8.97i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (6.68 - 3.86i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.32 - 3.32i)T - 83iT^{2} \)
89 \( 1 + (-8.91 - 15.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.100 + 0.100i)T + 97iT^{2} \)
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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

−8.754565413750359174644759369773, −8.505021931351398654232433966307, −7.73441560910007089510341173179, −6.64606671427454532708367674217, −5.72307681268356390046612581043, −5.15399493118883767344287935836, −3.86623919957779950740460105551, −3.59216497240637694147654931072, −2.11895725983231125121439351798, −0.850206035409223547229328943365, 1.32608466938909811822502114356, 2.00677624703499215856719307928, 3.22305429313778827272365030710, 4.47237670606012937416234558152, 4.95175347131009619033902862477, 6.18639844664700865582726583069, 6.90797978459403258310820934861, 7.47365379830176268165386292152, 8.447493022952814206762904567131, 9.087035994143396431580133221811

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