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Properties

Label 2-1200-15.8-c1-0-21
Degree $2$
Conductor $1200$
Sign $-0.927 + 0.374i$
Analytic cond. $9.58204$
Root an. cond. $3.09548$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.70 − 0.292i)3-s + (−2 + 2i)7-s + (2.82 + i)9-s + 5.65i·11-s + (−2.82 − 2.82i)17-s − 4i·19-s + (4 − 2.82i)21-s + (4.24 − 4.24i)23-s + (−4.53 − 2.53i)27-s + 5.65·29-s − 8·31-s + (1.65 − 9.65i)33-s + (−8 + 8i)37-s − 5.65i·41-s + (−2 − 2i)43-s + ⋯
L(s)  = 1  + (−0.985 − 0.169i)3-s + (−0.755 + 0.755i)7-s + (0.942 + 0.333i)9-s + 1.70i·11-s + (−0.685 − 0.685i)17-s − 0.917i·19-s + (0.872 − 0.617i)21-s + (0.884 − 0.884i)23-s + (−0.872 − 0.487i)27-s + 1.05·29-s − 1.43·31-s + (0.288 − 1.68i)33-s + (−1.31 + 1.31i)37-s − 0.883i·41-s + (−0.304 − 0.304i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.927 + 0.374i$
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: \(1\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ -0.927 + 0.374i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 + 0.292i)T \)
5 \( 1 \)
good7 \( 1 + (2 - 2i)T - 7iT^{2} \)
11 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (2.82 + 2.82i)T + 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (-4.24 + 4.24i)T - 23iT^{2} \)
29 \( 1 - 5.65T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + (8 - 8i)T - 37iT^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + (2 + 2i)T + 43iT^{2} \)
47 \( 1 + (1.41 + 1.41i)T + 47iT^{2} \)
53 \( 1 + (5.65 - 5.65i)T - 53iT^{2} \)
59 \( 1 + 5.65T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (6 - 6i)T - 67iT^{2} \)
71 \( 1 + 11.3iT - 71T^{2} \)
73 \( 1 + (8 + 8i)T + 73iT^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + (-9.89 + 9.89i)T - 83iT^{2} \)
89 \( 1 + 11.3T + 89T^{2} \)
97 \( 1 + (-8 + 8i)T - 97iT^{2} \)
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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

−9.402822132059400201473778505126, −8.828958013841248821221912085448, −7.33141066806559101920320673994, −6.90312460301925017665630973448, −6.14338251429471009493355995330, −4.95349925205421325153156435037, −4.60051738945155253587241994299, −2.95623639703052422194080173491, −1.83313232375084836687615780898, 0, 1.33915074520200397122627638469, 3.32330537809552337108992570988, 3.91058196048793774274400447395, 5.16720725357775466167810705883, 5.98950487083743241321939682913, 6.59456570505574244109479947746, 7.47806463586219298492644490378, 8.548203821529180701599916673453, 9.408721572497026155637185590260

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