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Properties

Label 2-2100-35.13-c1-0-20
Degree $2$
Conductor $2100$
Sign $-0.746 + 0.664i$
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  + (0.707 − 0.707i)3-s + (−1.51 + 2.16i)7-s − 1.00i·9-s + 1.10·11-s + (−3.52 + 3.52i)13-s + (−3.23 − 3.23i)17-s + 0.916·19-s + (0.457 + 2.60i)21-s + (−4.43 − 4.43i)23-s + (−0.707 − 0.707i)27-s − 8.28i·29-s − 4.59i·31-s + (0.781 − 0.781i)33-s + (2.30 − 2.30i)37-s + 4.98i·39-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.574 + 0.818i)7-s − 0.333i·9-s + 0.333·11-s + (−0.977 + 0.977i)13-s + (−0.785 − 0.785i)17-s + 0.210·19-s + (0.0997 + 0.568i)21-s + (−0.925 − 0.925i)23-s + (−0.136 − 0.136i)27-s − 1.53i·29-s − 0.825i·31-s + (0.136 − 0.136i)33-s + (0.378 − 0.378i)37-s + 0.797i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.746 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.746 + 0.664i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1693, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.746 + 0.664i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7390450155\)
\(L(\frac12)\) \(\approx\) \(0.7390450155\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 \)
7 \( 1 + (1.51 - 2.16i)T \)
good11 \( 1 - 1.10T + 11T^{2} \)
13 \( 1 + (3.52 - 3.52i)T - 13iT^{2} \)
17 \( 1 + (3.23 + 3.23i)T + 17iT^{2} \)
19 \( 1 - 0.916T + 19T^{2} \)
23 \( 1 + (4.43 + 4.43i)T + 23iT^{2} \)
29 \( 1 + 8.28iT - 29T^{2} \)
31 \( 1 + 4.59iT - 31T^{2} \)
37 \( 1 + (-2.30 + 2.30i)T - 37iT^{2} \)
41 \( 1 + 3.15iT - 41T^{2} \)
43 \( 1 + (2.70 + 2.70i)T + 43iT^{2} \)
47 \( 1 + (4.71 + 4.71i)T + 47iT^{2} \)
53 \( 1 + (-6.41 - 6.41i)T + 53iT^{2} \)
59 \( 1 + 4.36T + 59T^{2} \)
61 \( 1 + 10.8iT - 61T^{2} \)
67 \( 1 + (-6.81 + 6.81i)T - 67iT^{2} \)
71 \( 1 - 4.79T + 71T^{2} \)
73 \( 1 + (9.72 - 9.72i)T - 73iT^{2} \)
79 \( 1 - 12.5iT - 79T^{2} \)
83 \( 1 + (0.227 - 0.227i)T - 83iT^{2} \)
89 \( 1 + 8.93T + 89T^{2} \)
97 \( 1 + (12.4 + 12.4i)T + 97iT^{2} \)
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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

−8.840974674594925235479414207359, −8.106437321087281155753943108763, −7.15213896305999552377228639132, −6.55512187873414231953812981925, −5.79291576803936017882791340198, −4.67576477648716422354868141959, −3.85405542148605917880989933467, −2.52721636388879213550368094329, −2.13931663302212707458673485524, −0.23276994174759461422714743330, 1.47476782101577519405721891112, 2.86428998441513781928070413889, 3.58211605195652322127913049576, 4.45531994552397095233716247783, 5.32324293328482589609920986757, 6.34255036131078780215214673849, 7.15660712490658524592110647920, 7.84620294068379223814498080937, 8.657691681033508849592526750746, 9.503099480126121858777517379165

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