Properties

Label 2-1800-120.107-c0-0-3
Degree $2$
Conductor $1800$
Sign $0.662 - 0.749i$
Analytic cond. $0.898317$
Root an. cond. $0.947795$
Motivic weight $0$
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)2-s + 1.00i·4-s + (1 − i)7-s + (−0.707 + 0.707i)8-s − 1.41i·11-s + (1 + i)13-s + 1.41·14-s − 1.00·16-s + (1.00 − 1.00i)22-s + 1.41i·26-s + (1.00 + 1.00i)28-s + (−0.707 − 0.707i)32-s + (−1 + i)37-s − 1.41i·41-s + 1.41·44-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + 1.00i·4-s + (1 − i)7-s + (−0.707 + 0.707i)8-s − 1.41i·11-s + (1 + i)13-s + 1.41·14-s − 1.00·16-s + (1.00 − 1.00i)22-s + 1.41i·26-s + (1.00 + 1.00i)28-s + (−0.707 − 0.707i)32-s + (−1 + i)37-s − 1.41i·41-s + 1.41·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.662 - 0.749i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ 0.662 - 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.823883313\)
\(L(\frac12)\) \(\approx\) \(1.823883313\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-1 + i)T - iT^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 + (-1 - i)T + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (1 - i)T - iT^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
53 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + 1.41T + T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.208116113098935324590202048879, −8.576300286080600876179035433467, −7.918280237630170996361649557032, −7.15402655892983036841783193052, −6.31269381469828824786763980612, −5.62105468480300655623066643085, −4.56295965746085436971958449694, −3.97170366370086445782045915606, −3.05662780229486034017187702195, −1.44817319801647164540447848030, 1.52902223660804451088843312210, 2.32188695783414856389982357589, 3.41868319869982043926084036739, 4.47218056618065224555619094941, 5.20080844306601801970805807940, 5.81681196100221932068276425534, 6.82163482035063539648652382103, 7.88109279715263019649285873486, 8.682436847002515711980900592469, 9.475538939175700840627812522786

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