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Properties

Label 2-2100-105.89-c1-0-7
Degree $2$
Conductor $2100$
Sign $-0.0778 - 0.996i$
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  + (−1.42 + 0.990i)3-s + (−0.156 − 2.64i)7-s + (1.03 − 2.81i)9-s + (−4.92 + 2.84i)11-s + 1.43·13-s + (2.81 − 1.62i)17-s + (−5.43 − 3.13i)19-s + (2.83 + 3.59i)21-s + (0.510 − 0.884i)23-s + (1.31 + 5.02i)27-s + 4.95i·29-s + (−2.28 + 1.31i)31-s + (4.17 − 8.91i)33-s + (7.88 + 4.55i)37-s + (−2.04 + 1.42i)39-s + ⋯
L(s)  = 1  + (−0.820 + 0.572i)3-s + (−0.0592 − 0.998i)7-s + (0.345 − 0.938i)9-s + (−1.48 + 0.857i)11-s + 0.398·13-s + (0.682 − 0.393i)17-s + (−1.24 − 0.719i)19-s + (0.619 + 0.784i)21-s + (0.106 − 0.184i)23-s + (0.253 + 0.967i)27-s + 0.920i·29-s + (−0.410 + 0.236i)31-s + (0.727 − 1.55i)33-s + (1.29 + 0.748i)37-s + (−0.326 + 0.227i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0778 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0778 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.0778 - 0.996i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.0778 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7406544803\)
\(L(\frac12)\) \(\approx\) \(0.7406544803\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.42 - 0.990i)T \)
5 \( 1 \)
7 \( 1 + (0.156 + 2.64i)T \)
good11 \( 1 + (4.92 - 2.84i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.43T + 13T^{2} \)
17 \( 1 + (-2.81 + 1.62i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.43 + 3.13i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.510 + 0.884i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.95iT - 29T^{2} \)
31 \( 1 + (2.28 - 1.31i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.88 - 4.55i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.203T + 41T^{2} \)
43 \( 1 - 3.91iT - 43T^{2} \)
47 \( 1 + (-9.98 - 5.76i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.32 + 9.21i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.739 - 1.28i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.62 - 4.40i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.05 - 2.34i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.08iT - 71T^{2} \)
73 \( 1 + (-4.51 - 7.82i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.80 + 10.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.1iT - 83T^{2} \)
89 \( 1 + (7.53 - 13.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−9.619285144652097186081578051172, −8.538149316566784285097787109032, −7.60352648957123064717283248646, −6.96102718463278784155831248658, −6.14871776533945091029562978778, −5.10857253305357527918354683230, −4.62988617463757008897821905429, −3.72576119549393995990353097496, −2.58509122175583530750877129807, −0.978346832761060334907419578130, 0.35332475009577076665728560355, 1.91439153488282405115853578302, 2.75866416574144390912284186490, 4.04950513036159387515754593310, 5.22773601039923647319552636161, 5.86650932889700767762556060985, 6.16942502427691123604045222207, 7.46369353078077746457656140039, 8.086354093695664011598630389639, 8.660050832050527701747248785824

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