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Properties

Label 2-1800-120.107-c0-0-0
Degree $2$
Conductor $1800$
Sign $-0.927 - 0.374i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.927 - 0.374i$
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: \((2,\ 1800,\ (\ :0),\ -0.927 - 0.374i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.167551670\)
\(L(\frac12)\) \(\approx\) \(1.167551670\)
\(L(1)\) 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

−9.520204590547621071716650991709, −9.144545145487992472420881193564, −7.895989800215647186663572721488, −7.44772763467645544458786018300, −6.49901750258204395498512771592, −5.82589641498245419766280779664, −5.02223090578690590049741172893, −4.21728945187121086787354569003, −2.96381764458948259670638488428, −2.40408703140675698019818526721, 0.66146810682819491308341773297, 2.23409957568932189465353875614, 3.31717641019146120471707082414, 3.90871675579690642875962784333, 4.86551126754225858659364006929, 5.85559088071455843949990245700, 6.63083487974213064590274970000, 7.24110512202807194213351332764, 8.534166688336413412144830788626, 9.401138983101710837516205855765

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