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Properties

Label 2-2100-15.8-c1-0-25
Degree $2$
Conductor $2100$
Sign $0.858 + 0.512i$
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  + (1.65 + 0.522i)3-s + (0.707 − 0.707i)7-s + (2.45 + 1.72i)9-s − 3.21i·11-s + (−3.87 − 3.87i)13-s + (4.34 + 4.34i)17-s − 6.27i·19-s + (1.53 − 0.797i)21-s + (−2.64 + 2.64i)23-s + (3.14 + 4.13i)27-s + 8.25·29-s + 2.30·31-s + (1.67 − 5.30i)33-s + (5.79 − 5.79i)37-s + (−4.36 − 8.41i)39-s + ⋯
L(s)  = 1  + (0.953 + 0.301i)3-s + (0.267 − 0.267i)7-s + (0.817 + 0.575i)9-s − 0.968i·11-s + (−1.07 − 1.07i)13-s + (1.05 + 1.05i)17-s − 1.44i·19-s + (0.335 − 0.174i)21-s + (−0.550 + 0.550i)23-s + (0.605 + 0.795i)27-s + 1.53·29-s + 0.414·31-s + (0.292 − 0.922i)33-s + (0.952 − 0.952i)37-s + (−0.699 − 1.34i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 + 0.512i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.858 + 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.858 + 0.512i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.858 + 0.512i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.570379166\)
\(L(\frac12)\) \(\approx\) \(2.570379166\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.65 - 0.522i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
good11 \( 1 + 3.21iT - 11T^{2} \)
13 \( 1 + (3.87 + 3.87i)T + 13iT^{2} \)
17 \( 1 + (-4.34 - 4.34i)T + 17iT^{2} \)
19 \( 1 + 6.27iT - 19T^{2} \)
23 \( 1 + (2.64 - 2.64i)T - 23iT^{2} \)
29 \( 1 - 8.25T + 29T^{2} \)
31 \( 1 - 2.30T + 31T^{2} \)
37 \( 1 + (-5.79 + 5.79i)T - 37iT^{2} \)
41 \( 1 + 9.59iT - 41T^{2} \)
43 \( 1 + (3.88 + 3.88i)T + 43iT^{2} \)
47 \( 1 + (-1.57 - 1.57i)T + 47iT^{2} \)
53 \( 1 + (2.48 - 2.48i)T - 53iT^{2} \)
59 \( 1 - 0.317T + 59T^{2} \)
61 \( 1 + 3.94T + 61T^{2} \)
67 \( 1 + (-4.21 + 4.21i)T - 67iT^{2} \)
71 \( 1 - 9.52iT - 71T^{2} \)
73 \( 1 + (-1.46 - 1.46i)T + 73iT^{2} \)
79 \( 1 - 12.9iT - 79T^{2} \)
83 \( 1 + (-9.41 + 9.41i)T - 83iT^{2} \)
89 \( 1 - 4.62T + 89T^{2} \)
97 \( 1 + (5.42 - 5.42i)T - 97iT^{2} \)
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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

−8.964763859624892210846677829870, −8.179339599353835743527163917931, −7.76405469944471232142781405914, −6.89347115705239495070150599749, −5.72911111429665342811455233253, −4.97020954315725637999365944725, −3.98676820597683016488679506762, −3.13783766096170511420367302660, −2.36279534397924317747962543576, −0.873537005925984952100916035595, 1.37348708226364902271668263163, 2.32479981693808423731799124645, 3.13882143650497308419440173658, 4.40077556585818768403693846629, 4.85979005404541837941521750940, 6.26780031419420434242181586506, 6.92602690601335573728806123270, 7.86693489906784958919807221694, 8.122900245632237837859107065702, 9.307923718533209247567913399278

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