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Properties

Label 2-2100-21.5-c1-0-25
Degree $2$
Conductor $2100$
Sign $0.993 + 0.116i$
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  + (1.16 − 1.28i)3-s + (1.80 + 1.93i)7-s + (−0.284 − 2.98i)9-s + (−4.05 + 2.34i)11-s − 2.18i·13-s + (3.74 + 6.49i)17-s + (−0.638 − 0.368i)19-s + (4.58 − 0.0477i)21-s + (6.99 + 4.03i)23-s + (−4.15 − 3.11i)27-s − 1.15i·29-s + (8.95 − 5.16i)31-s + (−1.72 + 7.92i)33-s + (−2.30 + 3.99i)37-s + (−2.80 − 2.55i)39-s + ⋯
L(s)  = 1  + (0.672 − 0.739i)3-s + (0.680 + 0.732i)7-s + (−0.0947 − 0.995i)9-s + (−1.22 + 0.706i)11-s − 0.607i·13-s + (0.909 + 1.57i)17-s + (−0.146 − 0.0845i)19-s + (0.999 − 0.0104i)21-s + (1.45 + 0.842i)23-s + (−0.800 − 0.599i)27-s − 0.214i·29-s + (1.60 − 0.928i)31-s + (−0.300 + 1.37i)33-s + (−0.379 + 0.657i)37-s + (−0.449 − 0.408i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.993 + 0.116i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.993 + 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.383818865\)
\(L(\frac12)\) \(\approx\) \(2.383818865\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.16 + 1.28i)T \)
5 \( 1 \)
7 \( 1 + (-1.80 - 1.93i)T \)
good11 \( 1 + (4.05 - 2.34i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.18iT - 13T^{2} \)
17 \( 1 + (-3.74 - 6.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.638 + 0.368i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.99 - 4.03i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.15iT - 29T^{2} \)
31 \( 1 + (-8.95 + 5.16i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.30 - 3.99i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.43T + 41T^{2} \)
43 \( 1 - 9.24T + 43T^{2} \)
47 \( 1 + (-4.34 + 7.52i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.03 + 4.06i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.48 + 6.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.13 - 2.96i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.691 + 1.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.26iT - 71T^{2} \)
73 \( 1 + (-0.211 + 0.122i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.79 - 10.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16.4T + 83T^{2} \)
89 \( 1 + (0.658 - 1.14i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.84iT - 97T^{2} \)
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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

−8.842797585985703584688712574400, −8.148797364132031792923384698280, −7.79359743896922264389523421076, −6.93143217501794308425346032936, −5.82842072132224492451849781745, −5.29105589827410496440978303343, −4.10751587533461252020484437566, −2.94146553488230531354203487022, −2.26221997484949477555477784321, −1.15413492647051049010179299560, 0.934349205220245149337201031230, 2.58027628156395078705437551006, 3.12900720538704315368915543583, 4.38719381591646444075968278401, 4.88246356650618135662001292090, 5.70129140558107170472743231489, 7.13733542708561288927353407583, 7.57563353815831781092200286054, 8.465369279120549447410249680039, 9.007184664656412713079808500232

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