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Properties

Label 2-1800-8.5-c1-0-37
Degree $2$
Conductor $1800$
Sign $0.883 + 0.467i$
Analytic cond. $14.3730$
Root an. cond. $3.79118$
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  + (−0.5 − 1.32i)2-s + (−1.50 + 1.32i)4-s + 4·7-s + (2.50 + 1.32i)8-s + 2.64i·11-s + (−2 − 5.29i)14-s + (0.500 − 3.96i)16-s + 3·17-s − 2.64i·19-s + (3.50 − 1.32i)22-s − 4·23-s + (−6.00 + 5.29i)28-s + 4·31-s + (−5.50 + 1.32i)32-s + (−1.5 − 3.96i)34-s + ⋯
L(s)  = 1  + (−0.353 − 0.935i)2-s + (−0.750 + 0.661i)4-s + 1.51·7-s + (0.883 + 0.467i)8-s + 0.797i·11-s + (−0.534 − 1.41i)14-s + (0.125 − 0.992i)16-s + 0.727·17-s − 0.606i·19-s + (0.746 − 0.282i)22-s − 0.834·23-s + (−1.13 + 0.999i)28-s + 0.718·31-s + (−0.972 + 0.233i)32-s + (−0.257 − 0.680i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.883 + 0.467i$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ 0.883 + 0.467i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.620343642\)
\(L(\frac12)\) \(\approx\) \(1.620343642\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 1.32i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 4T + 7T^{2} \)
11 \( 1 - 2.64iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + 2.64iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 10.5iT - 37T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 - 5.29iT - 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 10.5iT - 53T^{2} \)
59 \( 1 - 5.29iT - 59T^{2} \)
61 \( 1 + 10.5iT - 61T^{2} \)
67 \( 1 + 7.93iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 7.93iT - 83T^{2} \)
89 \( 1 - T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−9.342930894559459406408521757719, −8.405045005268332595061343463689, −7.891761224491592367418943115529, −7.19567614153608720541570870813, −5.80693830481470919587157517997, −4.68047822305579603412968703388, −4.41887086618798079696856846938, −3.04376282399055491213764434496, −2.02778098476886713695472315312, −1.12823060393836815639798249686, 0.864400941770931342388050603032, 2.03823212514795102040794439823, 3.74876205364605684072413974342, 4.56928352908856870307360491855, 5.57268789145946460539453832887, 5.92291078850693285273149339803, 7.21037025497015431478106727053, 7.80303282432126757751807393390, 8.415269179480392202928099403977, 9.012798453309982750999777294457

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