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Properties

Label 2-1200-15.8-c1-0-13
Degree $2$
Conductor $1200$
Sign $0.391 - 0.920i$
Analytic cond. $9.58204$
Root an. cond. $3.09548$
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 + 1.41i)3-s + (0.414 − 0.414i)7-s + (−1.00 + 2.82i)9-s − 4.82i·11-s + (1.82 + 1.82i)13-s + (3.82 + 3.82i)17-s + 4.82i·19-s + (1 + 0.171i)21-s + (−1.58 + 1.58i)23-s + (−5.00 + 1.41i)27-s + 7.65·29-s + 5.65·31-s + (6.82 − 4.82i)33-s + (0.171 − 0.171i)37-s + (−0.757 + 4.41i)39-s + ⋯
L(s)  = 1  + (0.577 + 0.816i)3-s + (0.156 − 0.156i)7-s + (−0.333 + 0.942i)9-s − 1.45i·11-s + (0.507 + 0.507i)13-s + (0.928 + 0.928i)17-s + 1.10i·19-s + (0.218 + 0.0374i)21-s + (−0.330 + 0.330i)23-s + (−0.962 + 0.272i)27-s + 1.42·29-s + 1.01·31-s + (1.18 − 0.840i)33-s + (0.0282 − 0.0282i)37-s + (−0.121 + 0.706i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.391 - 0.920i$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ 0.391 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.068848291\)
\(L(\frac12)\) \(\approx\) \(2.068848291\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.41i)T \)
5 \( 1 \)
good7 \( 1 + (-0.414 + 0.414i)T - 7iT^{2} \)
11 \( 1 + 4.82iT - 11T^{2} \)
13 \( 1 + (-1.82 - 1.82i)T + 13iT^{2} \)
17 \( 1 + (-3.82 - 3.82i)T + 17iT^{2} \)
19 \( 1 - 4.82iT - 19T^{2} \)
23 \( 1 + (1.58 - 1.58i)T - 23iT^{2} \)
29 \( 1 - 7.65T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + (-0.171 + 0.171i)T - 37iT^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + (-2.41 - 2.41i)T + 43iT^{2} \)
47 \( 1 + (6.41 + 6.41i)T + 47iT^{2} \)
53 \( 1 + (3 - 3i)T - 53iT^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 + (4.07 - 4.07i)T - 67iT^{2} \)
71 \( 1 + 6.48iT - 71T^{2} \)
73 \( 1 + (6.65 + 6.65i)T + 73iT^{2} \)
79 \( 1 + 4.82iT - 79T^{2} \)
83 \( 1 + (-5.24 + 5.24i)T - 83iT^{2} \)
89 \( 1 + 4.34T + 89T^{2} \)
97 \( 1 + (1 - i)T - 97iT^{2} \)
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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

−10.02642121291734663248837509184, −9.009534693268027146235233074958, −8.224238252678064061040007730513, −7.917667034539334198670479912149, −6.33564638108919453824947338459, −5.73737725336631536026890676842, −4.57654636027014894737182158983, −3.68193822661456719619718423470, −2.98750177986397782142118324957, −1.41800298114524329292012137314, 0.942111914166941289948394546811, 2.27278774160508165359373902784, 3.08923233077835281193056801137, 4.40663893044425551023444704375, 5.35142498583258395507090100484, 6.54481875889106710777271262470, 7.12755054935000369346879254236, 7.962027500137919104997783376760, 8.633109806568622370130902690208, 9.590698655881336011871324123379

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