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Properties

Label 2-2100-35.17-c1-0-3
Degree $2$
Conductor $2100$
Sign $-0.477 - 0.878i$
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  + (0.965 − 0.258i)3-s + (−2.08 + 1.62i)7-s + (0.866 − 0.499i)9-s + (0.293 − 0.507i)11-s + (−3.67 − 3.67i)13-s + (1.12 + 4.21i)17-s + (3.38 + 5.86i)19-s + (−1.59 + 2.11i)21-s + (−8.61 − 2.30i)23-s + (0.707 − 0.707i)27-s + 3.32i·29-s + (1.01 + 0.585i)31-s + (0.151 − 0.566i)33-s + (−1.89 + 7.06i)37-s + (−4.50 − 2.59i)39-s + ⋯
L(s)  = 1  + (0.557 − 0.149i)3-s + (−0.788 + 0.615i)7-s + (0.288 − 0.166i)9-s + (0.0883 − 0.153i)11-s + (−1.01 − 1.01i)13-s + (0.273 + 1.02i)17-s + (0.776 + 1.34i)19-s + (−0.347 + 0.460i)21-s + (−1.79 − 0.481i)23-s + (0.136 − 0.136i)27-s + 0.617i·29-s + (0.182 + 0.105i)31-s + (0.0264 − 0.0985i)33-s + (−0.311 + 1.16i)37-s + (−0.720 − 0.416i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.477 - 0.878i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.477 - 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9944787482\)
\(L(\frac12)\) \(\approx\) \(0.9944787482\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 \)
7 \( 1 + (2.08 - 1.62i)T \)
good11 \( 1 + (-0.293 + 0.507i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.67 + 3.67i)T + 13iT^{2} \)
17 \( 1 + (-1.12 - 4.21i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.38 - 5.86i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (8.61 + 2.30i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 3.32iT - 29T^{2} \)
31 \( 1 + (-1.01 - 0.585i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.89 - 7.06i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 3.44iT - 41T^{2} \)
43 \( 1 + (-0.0439 + 0.0439i)T - 43iT^{2} \)
47 \( 1 + (-0.731 - 0.195i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.05 - 11.3i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.90 - 5.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.44 - 2.56i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.3 - 3.05i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.58T + 71T^{2} \)
73 \( 1 + (-9.30 + 2.49i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (11.8 - 6.83i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (10.6 + 10.6i)T + 83iT^{2} \)
89 \( 1 + (2.36 + 4.09i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.70 - 3.70i)T - 97iT^{2} \)
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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

−9.412265369264209952365405704864, −8.418480018391934471672960396567, −7.981148155170202792042912716722, −7.14073484994965478911524790538, −6.03117860587956620192204654544, −5.67358131084356807657118369367, −4.36949885963556046769914599078, −3.37521798567661411106857207056, −2.71966492164208141147320503505, −1.52461905384477509095388492142, 0.30783299747262780929273878330, 2.01963965688132805762717860076, 2.92482423200690348648208754303, 3.91364743050552153948204296062, 4.62650645274372484679211750160, 5.62758324313927261014442480412, 6.82107050843122328650924286052, 7.20793693190701396028216015407, 7.965399757586002955519826183181, 9.122073799725258659206969177442

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